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Design optimization procedure using robustness for minimizing transmission error in spur and helical gears

Sundaresan, Sivakumar, Ph.D. The Ohio State University, 1992

UMI 300N.ZeebRA Ann Arbor, MI 48106

DESIGN OPTIMIZATION PROCEDURE USING ROBUSTNESS FOR MINIMIZING TRANSMISSION ERROR IN SPUR AND HELICAL GEARS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Doctor of Philosophy in the Graduate School of the Ohio State University

By Sivakumar Simdaresan, B.Tech., M.S.

***** The Ohio State University 1992

Approved by Dissertation Committee

I wish to thank my advisers, Dr. Kosuke Ishii and Dr. Donald R. Houser for their invaluable guidance and assistance throughout the course of this research. Not only did they act as mentors, directing my work, they created a congenial atmosphere for development and exchange of ideas.

In addition I would like to express my gratitude to the other members of the dessertation commitee, Dr. Gary L. Kinzel and Dr. Henry R. Busby for their suggestions, review and evaluation of this thesis.

I extend my appreciation to the Graduate School and the sponsors of the Gear Dynamics and Gear Noise Research Laboratory for supporting this research activity financially. I acknowledge the students of the Gear Dynamics and Gear Noise Research Laboratory and Life Cycle Engineering Group at Ohio State (LEGOS) for their helpful suggestions during the course of this work.

Finally, I thank my wife, Latha and my parents for their support and encouragement.

iii VITA

January 2,1966 Bom - Madurai, Tamil Nadu, India.

06/1987 Bachelor of Technology, Indian Institute of Technology, Madras, India.

07/87 -12/87 Graduate Trainee Engineer, Hindustan Motors Limited, Hosur, India.

01/88 - 06/88 Graduate Teaching Associate, Dept. of Mechanical Engineering, The Ohio State University, Columbus, Ohio.

06/88 - 06/91 Graduate Research Associate, Gear Dynamics and Gear Noise Research Lab. The Ohio State University, Columbus, Ohio.

07/91 -12/91 University Presidential Fellow The Ohio State University, Columbus, Ohio.

01/92 - 06/92 College Intern Core Engineering Center of Integrated Products Eastman Kodak Company, Rochester, New York

07/92 - Present University Presidential Fellow The Ohio State University, Columbus, Ohio.

IV PUBLICATIONS

1) Sundaresan. S.. Ishii, K., and Houser, D.R. (1991). A Procedure using Manufacturing Variance to Design Gears with Minimum Transmission Error, Journal Of Mechanical Design, Transactions of the ASME, September 1991, Vol.113, pp 318-324.

2) Sundaresan. S.. Ishii, K., and Houser, D.R. (1991). Design Optimization for Robustness using Performance Simulation Programs, Proceedings of the the American Society of Mechanical Engineers Design Automation Conference, DE-Vol. 32-1, pp 249-256, Miami, Florida, Sept 22-25. Also to appear in Journal of Engineering Optimization. 3) V. Ramamurti, H. Srinivasan, R. Sekhar, S.Sekar and S.Sivakumar (1989). Dynamic Analysis of Heavy Duty Wheels, Journal of Computers and Structures, Vol 31, No. 2 pp 275-285. 4) Sundaresan. S.. Ishii, K„ and Houser, D.R. (1990). A Procedure that Accounts for Manufacturing Errors in the Design Minimization of Transmission Error in Helical Gears, American Gear Manufacturers Association(AGMA) Fall Technical Meeting, Toronto, Canada. 5) Sundaresan. S.. Ishii, K., and Houser, D.R. (1991). Design of Helical Gears with Minimum Transmission Error under Manufacturing and Operational Variances, Proceedings of the Japan Society of Mechanical Engineers International Conference on Motion and Power Transmissions, Hiroshima, Japan, Nov 24-26,1991. 6) Bhakuni, N., Ishii, K., Tragesser, A. and Sundaresan. S. (1991). Structural Optimization methods for Aluminum Beverage Can Bottoms, Proceedings of the the American Society of Mechanical Engineers Design Automation Conference, DE-Vol. 32-1, pp 265-271, Miami, Florida, Sept 22-25. 7) Gear Design Program Overcomes Effects of Manufacturing Variations, edited by Joe Brown, Associate editor, Power Transmission Design, March 1991, pp 31-33, based on publications #1 and #4.

FIELDS OF STUDY

Major Field: Mechanical Engineering

v TABLE OF CONTENTS

DEDICATION ii ACKNOWLDEGEMENTS iii VITA iv TABLE OF CONTENTS vi LIST OF FIGURES ix LISTOFTABLES xii NOMENCLATURE xiii DISSERTATION ABSTRACT xvi

CHAPTER I INTRODUCTION 1 1.1 Motivation 1 1.2 Design For Robustness (DFR) 2 1.2.1 Concept of Statistical Optimum 3 1.2.2 Literature review in statistical optimization 4 1.3 Gear design with minimum transmission error 11 1.3.1 Background 11 1.3.2 Literature review 13 1.4 Outline 18

II MINIMIZING TRANSMISSION ERROR USING TOOTH MODIFICATIONS 20 2.1 Introduction 20 2.2 Profile and Lead Modifications 20 2.3 Cross modification 28 2.4 Topographical modifications 31 2.5 Effect of Misalignment and Torque Variations 33

vi 2.6 Summary 34

IB UNCONSTRAINED STATISTICAL OPTIMIZATION 36 3.1 Introduction 36 3.2 Generalized Problem 36 3.3 Introduction to Design of Experiments 38 3.4 Procedure 42 3.5 Helical Gear Example 44 3.6 Discussion of Results 47 3.7 Summary 49

IV CONSTRAINED STATISTICAL OPTIMIZATION 51 4.1 Introduction 52 4.2 Background 50 4.3 Literature Review 57 4.4 Procedure 59 4.4.1 Method 1: Brute Force with Heuristics 59 4.4.2 Method 2: Constraints with built-in constraint variation 61 4.4.3 Method 3: Approach using KKT conditions 62 4.4.4 Discussion 65 4.5 Spur Gear Example 66 4.6 Summary 69

V PARAMETRIC STUDY USING ILLUSTRATIVE EXAMPLES 70 5.1 Introduction 70 5.2 Example #1 70 5.3 Example # 2 76 5.2.1 Effect of Hob shift 77 5.2.2 Optimization example 82 5.3 Example # 3 85 5.5 Example # 4 88 5.6 Example # 5 91 5.7 Summary 96

VI CONCLUSIONS 98 6.1 Concluding Remarks 98

vu 6.2 Contributions of this Research .• 100 6.2.1 Contributions in statistical optimization 100 6.2.2 Contributions in gear design 101 6.3 Recommendations for Future Work 102

BIBLIOGRAPHY 105

APPENDICES A INTRODUCTION TO ANALYSIS OF VARIANCE 110 A.l Introduction 110 A.2 Illustration 110

B ALGORITHMS USED IN OPTIMIZATION SCHEME 115 B.l Introduction 115 B.2 Golden Section Method 115 B.3 BFGS Variable Metric Method 118 B.4 Augmented Lagrange Multiplier Method 120

vui LIST OF FIGURES Figure Page 1.1 Concept of statistical optimum 4 1.2 Possible design space for two variables 9 1.3 Profile specification chart 17 2.1 Profile modification 21 2.2 Lead modification 21 2.3 Contour plots of equal PPTE in microinches for various parabolic profile modification with a fixed lead modification of 0.0001" 23 2.4 Contour plots of equal PPTE in microinches for various parabolic profile modification with a fixed corner modification of 0.0001" 24 2.5 Normalized load distribution over the contact zone for optimum parabolic profile with a fixed lead modification of 0.0001" 25 2.6 Optimum profile modification 26 2.7 Load distribution factors for various parabolic profile modifications with a fixed lead modification of 0.0001" 27 2.8 Cross modification 28 2.9 Contour plots of equal PPTE in microinches for various parabolic cross modifiations with a fixed lead modification of 0.0001" 29 2.10 Load distribution factors for various parabolic cross modifications with a fixed lead modification of 0.0001" 30

ix 2.11 Normalized load distribution over the contact zone for optimum parabolic profile with a fixed lead modification of 0.0001" 31 2.12 Contour plot of equal tooth modification in microinches over the contact zone 32 2.13 Contours of equal PPTE in microinches for a gear mesh with optimum profile modification for various values of misalignment and torque 33 2.14 Contours of equal PPTE in microinches for a gear mesh with optimum cross modification for various values of misalignment and torque 34 3.1 Full and fractional factorial experiments 39 3.2 Optimization results 46 3.3 Worst case of L8 for peak and statistical optimum 46 4.1 Criticality of constraints 56 4.2 Constraints and contours of objective function 68 5.1 Constraints and contours of objective function 73 5.2 Contours of equal objective function value for optimum design in thefirst step with various profile and lead modifications 74 5.3 Costraints and contours of objective function for optimum profile modification in the second step 75 5.4 Contours of equal PPTE for various gear meshes defined by hob shift coefficient and pressure angle for 40 teeth in the pinion 78 5.5 Contours of equal PPTE for various gear meshes defined by hob shift coefficient and pressure angle for 30 teeth in the pinion 78 5.6 Transmission Error v/s number of mesh cycles for a LCR spur gear with various hob shift coefficients 79 5.7 Transmission Error v/s number of mesh cycles for a helical gear (FCR=1) with various hob shift coefficients 80

x Transmission Error v/s number of mesh cycles for a helical gear (FCR > 1.0) with various hob shift coefficients 81 Contours of PPTE in microinches for various spur gear meshes 93 Contours of PPTE in microinches for various helical gear meshes 94 Contours of PPTE in microinches for various helical gear meshes 95

xi LIST OF TABLES

Table Page 2.1 Gear Mesh Parameters 26 3.1 L8 Orthogonal Array 41 3.2 Gear Geometry 45 3.3 Comparison of peak and Statistical optimum 46 3.4 PPTE at all worst combinations for peak and statistical optima 48 3.5 Sums of Squares for peak and statistical optima 48 4.1 Results of statistical optimization using four diffemet methods 68 5.1 Gear Geometry 76 5.2 Gear Geometry for the Optimum 84 5.3 Gear Geometry for the Optimum 87 5.4 Gear Geometry for the target design 90 5.5 PPTE at all worst combinations 91 5.6 Sum of squares 91 A.l Illustrative Data Ill

xii NOMENCLATURE

(ab)ij Effect of interaction between factors A and B ai Effect of factor A at level i b Right hand side of the constraint equation F Statistical objective function f Performance function fc Performance at the target design fi Performance at the worst combination of design variables g Inequality constraint h Equality constraint L Lagrangian K Approximation to inverse of the Hessian matrix in BFGS method H Hessian matrix n Number of design variables

Np Number of teeth in the pinion p(x) Probability density function of variable x

Pnd Normal generating diametral pitch s.t. Such that S Solution space S/N Signal to Noise ratio T Tolerance space W Corner space

xiii x Design variable

X Vector of n design variables x+ Upper bound for x

X" Lower bound for x

*t Target design vector yi Performance values e Element of V For all

vx Gradient operator 3 There exists a Weighting factor P Lagrange multiplier for equality constraints p* Lagrange multiplier for equality constraints

Ab Variation in b Af Variation in f due to design variables

Ag Variation in g due to design variables Ax Variation in design variable x Ax Vector of variations in all variables

<|>n Normal generating pressure angle X Lagrange multiplier for inequality constraints

X* Optimum Lagrange multiplier

V Mean a Deviation a2 Variance

#FE Number of function evaluations #GE Number of constraint evaluations

XIV ALM Augmented Lagrange Multiplier BFGS Broydon-Fletcher-Goldfarb- Shanno variable metric method FCR Face Contact Ratio HCR High Contact Ratio (PCR > 2.0) HPDTC Highest Point of Double Tooth Contact HPSTC Highest Point of Single Tooth Contact LCR Low Contact Ratio (PCR < 2.0) LDP Load Distribution Program OA Orthogonal Array PCR Profile Contact Ratio PPTE Peak to Peak Transmission Error SAP Start radius of Active Profile SI Sensitivity Index TCR Total Contact Ratio

XV DESIGN OPTIMIZATION PROCEDURE USING ROBUSTNESS FOR MINIMIZING TRANSMISSION ERROR IN SPUR AND HELICAL GEARS

Sivakumar Sundaresan, Ph.D. The Ohio State University, 1992 Professor Kosuke Ishii, Co-adviser Professor Donald R. Houser, Co-adviser

This dissertation deals with the development of a procedure that incorporates manufacturing tolerances and operational variances in the design optimization stage to achieve designs with robust and optimal performance. The proposed procedure optimizes the expected value of a performance characteristic subject to a set of constraints and uses concepts from statistical design of experiments to approximate the expected value of a performance characteristic. The procedure incorporates uncertainties in design variables and variations in constraints due to uncertainty in design variables. This work discusses the following three methods to incorporate variations in constraints: 1) Method using heuristics that evaluates constraints at the worst combinations of design variables, 2) Method with built-in constraint variation that models constraints using first order Taylor expansion and 3) Method based on differentiating KKT optimality conditions. The methods developed in this dissertation enable engineers to design machine elements with optimum and robust performance. Design of spur and helical gears with minimum transmission error serves as the target application. A computer program called the Load

xvi Distribution Program (LDP) evaluates transmission error as a function of elastic properties of the gear mesh and errors in gear tooth profile. This dissertation characterizes transmission error by its peak to peak value. The two key gear design research issues addressed are: 1) Determination of optimal combination of geometric design variables like number of teeth, pressure angle that minimizes transmission error subject to constraints like minimum number of teeth to avoid undercut and maximum bending stress and 2) Finding the optimum tooth profile that not only minimizes transmission error but also is insensitive to manufacturing errors in tooth profile, shaft misalignment and torque variations. The statistical optimum shows considerable reduction in the sensitivity of transmission error to manufacturing errors, shaft misalignment and torque variation. This dissertation also identifies the effect of geometric design variables like number of teeth, pressure angle, helix angle, hob shift, diametral pitch and working depth on transmission error.

xvu CHAPTER I

INTRODUCTION

1.1 Motivation One of the important tasks in engineering design is to account for manufacturing errors and operational variances during the initial design stages. This work is aimed at developing a methodology to incorporate manufacturing and operational variances in the design optimization stage to achieve robust and optimal performance.

Traditionally, engineers conducted sensitivity analysis after design optimization. In recent years, researchers have developed methods to incorporate tolerances and variations during design stages. However, these methods are either customized towards a particular application or have been applied towards simple design problems with closed-form equations. Designers require a general methodology that incorporates variations in design variables as well as variations in constraints in the design optimization stage. This thesis addresses this need.

A simple solution for this problem is to use brute-force. Designers can consider all combinations of possible variations and determine the combination that exhibits the worst

1 2

performance. The worst combination is optimized during the design optimization stage. However, this approach is acceptable if there are fewer design variables or if the computer models are not time consuming. When the evaluation of performance is computationally expensive, one needs to adopt a probabilistic approach based on design of experiments. The methodology developed in this research uses concepts from statistical design of experiments during design optimization.

The proposed methodology will be applied to the design of spur and helical gears. Tooth surfaces of gears are modified from their ideal surfaces to minimize transmission error. Manufacturing errors in the profile, coupled with misalignment in the shafts on which gears are mounted, increase transmission error drastically. This research aims at studying the effects of tooth modifications on transmission error and its sensitivity to errors in manufacturing and operation. Additionally, this research studies the effect of geometric design variables such as number of teeth, pressure angle, etc. on transmission error.

The rest of this chapter is organized as follows: Section 1.2 discusses methods that are currently used in designing products that show uniform performance under uncertainties. Section 1.3 presents a literature review of the methods and models used in evaluating and minimizing transmission error in spur and helical gears. Section 1.4 gives the outline of the rest of the thesis.

1.2 Design For Robustness (DFR) This section deals with methods that are used to incorporate manufacturing and operational variances in the design optimization stage to achieve robust and optimal design. In conventional design process, a designer chooses values for design variables 3 such that performance is optimal. The sensitivity of an optimal design is measured after design optimization. However, if the manufacturing process associated with the design shows considerable variability, then the objective should be to select values for design variables such that design has optimal performance and at the same time is least sensitive to the enors associated with manufacturing.

One way to achieve robust and optimal performance is to apply statistical optimization techniques to the design process. In conventional optimization problems, the objective is to minimize/maximize a linear or a non-linear function of many variables subject to a set of constraints. In contrast, statistical optimization aims to optimize the expected value of a function subject to constraints with uncertainties. Each design variable is associated with a corresponding probability function. Section 1.2.1 explains the concept of statistical optimization and section 1.2.2 presents methods currently used to optimize the expected value of a performance characteristic.

1.2.1 Concept of Statistical Optimum. Figure 1.1 highlights the concepts behind statistical optimization. For illustrative purposes, we have used a function(f) of only one variable(x). Points L and M are the peak and statistical optimum respectively. If the target design is point L, a deviation 3a in the X could change function f to a value corresponding to point K. However, if point M is the target value, the change in function f is relatively small as function values at points L, M and N are nearly equal. Hence, the peak optimum may not be the best when one considers deviations of 3o which results in drastic deterioration of function f. When functions of two variables are considered, designers should be interested in flat surfaces near the peak optimum. Design Variable X

Fig. 1.1 Concept of Statistical optimum

1.2.2 Literature review in statistical optimization The statistical optimum in figure 1.1 could be achieved by optimizing both sensitivity and target performance. The concept of analyzing sensitivity is not new in engineering design. Traditionally, the sensitivity of an optimal design was measured after design optimization. Genichi Taguchi (1978) proposed a procedure based on design of experiments to account for performance variation due to errors in manufacturing and operation in the initial design stages. He categorized the design process into following three phases:

1) System design where initial values are assigned to design variables. 2) Parameter design where optimal values are determined by considering external influences like manufacturing errors. Kackar (1985) defines parameter design as "reducing variation in performance by reducing the sensitivity of an engineering design to sources of variation rather than controlling the sources." 3) Tolerance design where designers specify tolerances on how close the optimal values should be to achieve maximum performance.

Taguchi uses design of experiments concepts extensively in the parameter design stage and defines his measure of sensitivity as signal-to-noise ratio (S/N). In his method, he maximizes signal to noise ratio during the parameter design stage. The design variables are classified into two categories, namely, control factors and noise factors. The control factors are those that affect target performance but do not affect signal-to-noise ratio and the noise factors are those that affect only signal-to-noise ratio. After determining the levels of each design variable that maximizes the signal-to-noise ratio, Taguchi uses the control factors to set the mean performance at the target.

Taguchi has defined three signal to noise ratios based on the three objectives: Smaller the better, Nominal the best, and Larger the better, (i) Smaller-the-better For problems in which the objective is to minimize performance characteristics (e.g.. counting defects), the signal to noise ratio is

i n S/N=-log10 i=l (1-1)

(ii) Nominal-the-best For problems in which the objective is to have performance as close to the target as possible, 6

i n S/N=-log10 .1=1 (1.2) (iii) Larger-the-better For problems in which the objective is to maximize performance characteristics, the signal to noise ratio is

S/N=-log10 (1.3)

During Taguchi's parameter design stage, two orthogonal arrays are used: one to model noise factors (noise outer array) and one to model control factors (design inner array). A 2-level OA can be used to model noise factors with "1" referring to -ve variation and "2" referring to +ve variation in the noise factors. For each trial in the inner array, a trial in the outer anay is carried out and the performance is evaluated. The signal to noise ratio is calculated for each trial in the inner array. Values are chosen such that S/N is maximized. Barker (1986) applied this methodology to design of butterfly, a plastic part used in the carburetor of a lawn mower engine. He used a L9 outer anay and L27 inner anay for 6 three-level design variables.

Box (1988) studied the effectiveness of the three signal to noise ratios in modeling the variance in performance and suggested the application of a measure that used logarithmic transformation of the system response. He showed that 5 sets of 4 data points can give the same signal to noise ratio but can differ significantly in their distribution. One really needs to study the distribution of data points before using signal to noise ratios as an criterion for evaluation. The choice of the functions used are usually domain dependent. 7

The goal in this thesis is to come up with a function that can be easily evaluated when using time-consumingperformanc e simulation programs like LDP.

Taguchi's techniques were based on direct experimentation. Designers often use computer programs to evaluate performance characteristics instead of actual experiments. In such cases, non-linear optimization techniques are used to determine the best combination of design variables that give optimum performance. d'Entremont and Ragsdell (1988) developed a non-linear programming code that applies Taguchi's concepts to design optimization. Their goal was to minimize variability when the design is constrained to have target performance. While Taguchi's method uses control factors to adjust the performance to target, Ragsdell's approach uses a constraint during optimization to adjust the performance to target. Ragsdell and Tsai (1988) used 2-level (L4) and 3-level (L9) inner orthogonal anays to evaluate the gradient and Hessian of the objective function F using finite difference methods. The objective function was a squared deviation from the target evaluated using a outer orthogonal anay. In many design optimization problems, we do not have any target performance value nor are the mean and variance independent. As the mean changes, the standard deviation may change too and vice versa. One cannot minimize the standard deviation and then bring the mean to target. Minimizing both target performance and sensitivity is the aim and hence the above methods are unsuitable in such cases.

A measure equivalent to the Taguchi signal to noise ratio is f/Af where Af represents the variation in performance f. In many cases, the design achieved by optimizing target performance will be the least robust and trade-off has to be made between target performance and robustness. In such cases, one should ideally optimize the expected performance value of a design. For example, if the objective is to minimize function f(x) 8 with x having a probability p(x), then the objective function during statistical optimization would be

x+ F(x)= Jp(x)f(x)dx (1.4)

In equation (1.4), x+ and x" represent the upper and lower bounds for the design variable x. In cases where function f(x) is evaluated using computer programs, using equation (1.4) would be computationally time consuming. In most problems, p(x) is an unknown probability function. Our aim will be to approximate equation (1.4) by an expression that will be easier to evaluate using computer programs.

Sundaresan et al. (1989) evaluated the function at the worst combinations of the design variables and defined sensitivity index (SI) of a design as a root mean square value of the difference between the function values at the worst cases and the function value at the target design. For two variables xi and x2 with deviations a\ and a2, we considered 4 worst combinations. In figure 1.2, vertices 1 to 4 conespond to the 4 worst combinations. Equation 1.5 defines the sensitivity index. (fi fc)2 Hil ~ (1.5)

The objective function in statistical optimization was a weighted average of SI and function at the target fc.

F = ccfc +(l-a)SI 0£a

#$§&#** o2 mst x2 Untile 1 I W

xl K^-al H Fig. 1.2 Possible design space for two variables

To avoid penalizing fc, Sundaresan et al. (1989) recommended a to be greater than 0.5. When we consider variations in n design variables, the above technique requires 2n function evaluations to compute SI.

While this technique (equation 1.6) seems adequate for 2 design variables, the number of function evaluations increases exponentially for more than 2 variables. Hence, there is a need for reducing the number of function evaluations in statistical optimization. Chapter 3 will present a methodology that uses statistical design of experiments to reduce the number of function evaluations during optimization.

Sandgren (1989) formulated the statistical optimization problem as a non-linear goal programming problem with multiple objective functions. In this method, a priority (weight) is given to each objective function and constraint and procedure tries to satisfy as many goals it can satisfy starting with the most prefened one. This method addressed uncertainties through goal constraints. Objective function F in equation 1.6 can be regarded as a weighted average of two objective functions, namely target performance and sensitivity index. 10

However, in the methods discussed so far, we were considering the sensitivity of the objective function. In many design problems, the peak optimum could lie on one of the constraints. Statistical optimization procedures could result in an optimum that violates these constraints. In such cases, one would like to look into the sensitivity of the constraints along with the sensitivity of the objective function. Sandgren (1989) classified the constraints into two categories, namely, soft constraints which can be violated to an extent and hard constraints whose violation leads to an unacceptable design. The aim here is to achieve a robust optimum without any violated constraints.

One of the earliest forms of sensitivity analysis in non-linear programming was carried out using Lagrange multipliers. The Lagrange multipliers that satisfy Karush- Kuhn-Tucker conditions of optimality are sometimes called shadow prices in Operations Research. Lagrange multipliers give the change in the optimal function value for a unit change in the constraints.

A constrained statistical optimization problem can be formulated as

Minimize E[F(x)] subject to

gk(x)+Agk

°f A Af=I AXi i=1 3x" (1.8)

Agk=I ^Ax- (1.9) i=1 OX;

Chapter 4 deals with methods used to solve a non-linear statistical optimization problem involving variations in design variable and constraints. The aim is to not only minimize the effect of variations(tolerances) in design variables on performance but also to control the effect of variations in design variables on the active constraints. Use of Taguchi's method can help in studying the effect of variations in design variables on objective function and constraints. Chapter 4 presents a detailed discussion on the work reported in constrained statistical optimization and describes the methods used in this work.

1.3 Gear design with minimum transmission error 1.3.1 Background The design of gears has been a big challenge for mechanical engineers for a long time. Gear designers use certain rules of thumb and guidelines that they have developed over many years. One can view the process of gear design as a combination of two processes, namely, synthesis and analysis. The first step generates a set of feasible solutions that satisfy the design requirements and physical constraints. This procedure specifies the geometry of the gear and it's strength characteristics. The second step takes the feasible solutions and rates their performance for such issues as life, lubrication, noise, and vibration. This work combines the two steps and focuses on minimizing noise and vibration due to gear meshes. 12

The prediction of gear noise has always been a major concern in gear design. In recent years, researchers have established a relationship between static transmission enor and gear noise. Studies typically employ analytical tools to predict the static and dynamic transmission enors and subsequently, gear noise reduction leads to minimizing transmission enor.

The static transmission enor is the deviation in rotation from its ideal position when a gear pair is rotated with constant torque. Welbourn (1979) defined it as "the difference between the actual position of the output gear and the position it would occupy if the gear drive is perfect (infinite stiffness and conjugate teeth)." This enor is usually expressed in terms of the displacement along the line of conjugate gear action. The two main causes of the transmission enor at the meshing frequency and its harmonics are: 1) Non-ideal tooth profile and lead due to designed tooth modifications and manufacturing enors, 2) Elastic deflection of the gear tooth due to the transmitted load.

An existing program called the Load Distribution Program (LDP) evaluates the transmission enor as a function of elastic properties of the gear drive and tooth modifications and enors. The transmission enor is evaluated for discrete positions in the gear mesh cycle and this work characterizes the transmission enor in the following two ways: 1) Peak to peak transmission enor value(PPTE) and 2) Fourier amplitudes of the transmission enor at meshing frequency and its harmonics.

The explanation for the variation in the transmission enor and mesh stiffness lies in the conjugate action of gears. The number of pairs of teeth in contact for a low contact ratio spur gear pair oscillates between one and two. This leads to a sudden increase or 13 decrease in the mesh stiffness and hence, results in the variation of the transmission enor. One way to reduce this variation is to modify the profile of the gear tooth (by relieving the tip/root of the gear tooth) such that one tooth starts unloading when the second tooth makes initial contact. This process (called profile modification) also lessens the chance of premature gear tooth impact at the initiation of tooth contact.

However, this design procedure may produce lower contact ratios or substantial shortening of the path of contact when operated at lower loads. The designed modification is appropriate only for the designed load and will be detrimental when operated at lower loads. The manufacturing enor associated with machining this modification is of the same order as of the magnitude of the amount of material to be removed. Hence it is nearly impossible to achieve the designed modification and one can only use the designed modification as a target design. In addition to manufacturing enors and torque variations, the shafts on which the gears are mounted always have some degree of misalignment. The misalignment in the shafts causes a skewed load distribution which results in increased transmission enor. The gears are generally crowned (lead modification) to avoid the adverse effects of misalignment.

Hence there is a need for a procedure to design gears that have minimum transmission enor and, at the same time, are less sensitive to torque variations, shaft misalignment and enors in profile and lead modifications. Section 1.3.2 presents a brief literature review of the methods used in minimizing transmission error in spur and helical gears.

1.3.2 Literature review Optimization techniques have been used in many areas of gear design. Typical objective criteria include weight, space and load capacity. Errichello (1989) presented a 14 procedure for designing minimum weight spur and helical gear sets. He emphasized the presence of an optimum number of pinion teeth that maximizes the load capacity of the gear set. He considered surface fatigue and bending fatigue and determined an optimum number of pinion teeth as a ratio of pitting resistance to bending strength.

Some gear designs are limited by bearing size. Gear designers seek to find the optimum helix angle in helical gears such that the supporting bearings can withstand the axial thrust without substantial decrease in the load sharing of the gear mesh. In cases where center-distance is not a constraint, designers aim at the least center-distance to minimize space and pitch-line velocity. Reduction gears which require high bending strength usually have a pressure angle of 25°. The choice of the design variables such as diametral pitch, pressure angle, helix angle and number of teeth is based on the strength characteristics. Little work has been reported on the effect and sensitivity of these variables on transmission enor.

Modeling of the transmission enor started with simple spring models for spur gear teeth. Laskin (1968) approximated a tooth of a spur gear as a non-uniform cantilever beam with deflections due to bending, shear, and Hertzian stresses at the contact zone. Tavakoli and Houser (1986) presented a procedure similar to Box's complex method (Reklaitis et al., 1983) in order to determine the profile modification that resulted in minimized transmission enor. Conry and Seireg (1971) developed a simplex-type linear programming algorithm to evaluate load distribution between elastic bodies in contact. This algorithm was applied to gear teeth in contact to evaluate the transmission enor and load distribution. Conry and Seireg (1972) presented a technique to compute optimal modifications that provided uniform load distribution throughout the mesh cycle. LDP uses a tapered plate model for the gear tooth and adapts the algorithm presented by Conry 15 and Seireg. Yakubek (1985), and Yau (1987) improved upon this approach and modeled the gear tooth as a tapered plate with bending and shear deformations. Stegemiller (1989) modeled the base deflection of the gear teeth. Work done by Yakubek, Yau and Stegemiller have been incorporated in LDP.

Umezawa et. al. (1985) studied the vibrations of helical gears. They experimented with three categories of helical gears, namely, 1) Total contact ratio (TCR) less than 2.0, 2) TCR greater than 2.0 but face contact ratio (FCR) less than 1.0 and 3) TCR greater than 2.0 with FCR greater than 1.0. They report that gears belonging to the first category exhibit vibration characteristics similar to the spur gears. Category 2 gears show good performance when they undergo profile and lead modifications. Category 3 gears need not be profile modified as they perform quietly with unmodified profiles.

Rouverol and Pearce (1988) have projected the use of integer contact ratio gears to achieve constant mesh stiffness which shows substantial noise reduction over a range of loads. Integral contact ratio gears exhibit constant aggregate length of the tooth contact line. However, to achieve constant mesh stiffness, they report the need for balancing the tip strength of the pinion and the gear. Balancing the strength of the pinion and the gear can be achieved by varying the thickness of the tooth.

Lin (1989) studied the effect of profile modification on low contact ratio gears. He recommended parabolic tooth profile on gears that operate over a range of loads and linear tip modification on gears that operate at a constant load. He also concluded that excess modification is detrimental when gears are linearly modified. 16

Biggert and Houser (1988) presented a method for evaluating the sensitivity to enors in manufacturing and to operating at off-design loads of different profile modification designs for spur gears. They considered four design profiles and studied their sensitivity to deviations in the profile of the tooth and transmitted torque.

Umezawa (1989) presented performance diagrams which give the acceleration level of a helical gear pair with errors such as pitch enor, profile enor and lead enors. These diagrams present the non-dimensionalized acceleration level for any running speed by contour lines on the contact ratio domain defined by face contact ratio (FCR) on one axis and profile contact ratio (PCR) on the other. His diagrams are valid for all possible designs as mesh stiffness is non-dimensionalized and hence the mesh stiffness variation is only a function of FCR and PCR. Based on his results he reports that at low and medium speeds 1) Vibration levels are low for gears whose FCR > 1.0., 2) The influence of pressure angle enor on vibration levels is negligible. 3) Convex profile modification can reduce vibration. At higher speeds, convex profile reduces vibration except when PCR is an integer.

Simon (1989) presented developed a computer program similar to LDP to evaluate the optimal tooth tip relief and crowning for spur and helical gears that show uniform load distribution. He also studied the influence of gear geometry parameters on optimal tooth profile modification and derived using regression analysis an equation for calculating the optimal tooth tip relief.

Sundaresan et. al. (1989) considered the manufacturing enors associated with profile modification as a tolerance band around the target design modification. Figure 1.3 shows the profile specification chart with a simplified tolerance band. The true involute profile 17

is shown as a straight line and line PQ represents the target profile modification. Any curve within the tolerance band (shaded area) originating from a point on line AD and terminating in line BE represents a possible profile modification that could be manufactured. Their statistical optimization scheme used a weighted average of sensitivity index and transmission enor at the target design as the objective function.

TRUE INVOLUTE PROFILE

ROOT / A pp TIP ROLL ANGLE J^^^T

1 'RELIEF AMOUNT ^19 B Fig. 1.3 Profile specification chart

The methods presented so far dealt with minimizing transmission enor by determining optimum profile and lead modifications. However, these optimum modifications are optimum only for the design load. The load distribution will not be uniform if the shafts on which the gears are mounted are misaligned. Work has been reported on how torque variations and shaft misalignment affect transmission enor (Houser 1985, Biggert 1989, Lin 1989).

Rouverol (1991) developed a technique to design gear meshes that operate with zero transmission enor under a wide range of loads. By applying different profile modification at various locations along the face of the gear teeth, he obtained constant mesh stiffness at a wide range of torques. However, such designs are very sensitive to shaft misalignment which affect load distribution significantly. Hence, there is a need to 18 incorporate these operational variations in the design stage so that the designed gears have minimum transmission enor and at the same time are less sensitive to manufacturing enors, torque variations and shaft misalignment.

Another source of variation that affects transmission enor is the enor in pressure angle and helix angle associated with manufacturing. Injection molded plastic gears (Kleiss, 1991) and cast gears (Adjunta, 1991) show significant change in pressure angle and helix angle from that of the mold. The geometry and material properties of plastic gears undergo considerable change in geometry due to fluctuations in temperature and humidity. These changes significantly affect transmission enor and load distribution for such meshes. One needs to study the effect of geometric design variables on transmission enor and its sensitivity in order to design gear meshes that would show good performance at different environments. Chapter 5 addresses this need and studies the sensitivity of transmission enor to enors in pressure angle and helix angle.

1.4 Outline The rest of the thesis is organized as follows: Chapter 2 identifies with issues involved in minimizing transmission enor using three types of tooth modifications. Chapter 3 deals with unconstrained statistical optimization where the procedure uses concepts from statistical design of experiments to evaluate the expected value of a function. Chapter 4 presents three procedures for solving constrained statistical optimization problems that accomodate the variations in the design constraints due to the deviations in design variables. Chapter 5 applies the the techniques presented in chapters 3 and 4 to design spur and helical gears with minimum transmisssion enor. Chapter 5 also presents a brief discussion on the effect of geometric design variables like number of teeth, pressure angle, helix angle, diametral pitch, working depth and hob shift on 19 transmission enor. Chapter 6 concludes this thesis with the contributions made and recommendations for future work. CHAPTER H

MINIMIZING TRANSMISSION ERROR USING TOOTH MODIFICATIONS

2.1 Introduction One way to reduce peak to peak transmission enor is to modify the profile and lead of the gear teeth such that one mating tooth pair starts unloading when the second pair makes initial contact. This process called tooth modification also lessens the chance of premature gear tooth impact (comer contact) at the initiation of contact in spur and helical gears. Tooth modifications can be applied in the following three ways: 1) Conventional profile (involute) and lead modifications, 2) Cross modification and 3) 3-D topographical modification. This chapter discusses the merits and demerits of the three types of modifications and how they can be applied to achieve good load distribution and minimum transmission enor in spur and helical gears.

2.2 Profile and Lead Modifications This type of modification applies to spur and helical gears. Figures 2.1 and 2.2 show the profile and lead modifications respectively. Model used in LDP uses two parameters to characterize profile modification: 1) the start radius of profile modification and 2) the amount of relief at the tip/root of the gear teeth. In this model, the shape of the

20 21 modification can be either linear (linear profile modification) or quadratic (parabolic profile modification). Fig. 2.1 shows the profile of the gear tooth with radius scaled in terms of the roll angle. Line AB represents the true involute form of a gear tooth from its root to the tooth tip. Line PQ represents the modified tooth profile (linear modification) with point P and distance BQ conesponding to the starting roll angle of tooth modification and amount of tip relief respectively.

TRUE INVOLUTE FORM ROOT TIP

ROLL ANGLE

RELIEF AMOUNT Fig. 2.1 Profile modification.

Fig. 2.2 shows a typical parabolic lead modification. We characterize the lead modification by two parameters: 1) the starting position along the face width (point O) and 2) the amount of relief at the end faces of the gear tooth (distance AC and BD).

Fig. 2.2. Lead modification

The starting position of modification along the face width need not be at the center of the gear tooth. In wide face width gears, the starting position is usually in the first and 22

the last quarter of the face width and the central half of the gear face width is unmodified in the lead direction. Lead modifications are usually applied in spur gears to reduce the adverse effects of shaft misalignment. In helical gears, they can be combined with profile modification to effectively minimize transmission enor.

In low contact ratio spur gears, one can apply a relief amplitude at the tip of the gear tooth equal to the amount of tooth deflection when one pair of teeth is in contact, so that the mating teeth come into contact with low or no comer contacts. In the case of helical gears, this relief amplitude can be applied in either the involute profile or lead direction or both. One requires rules of thumb that will aid the designer in deciding the relief amounts to be applied in the lead and profile directions.

For low and high contact ratio spur gears, Sundaresan, et al. (1989) showed a three dimensional plot of the transmission enor as a function of starting roll angle of modification and amount of tip relief. Their study showed that there are many combinations of starting roll angle and relief amplitude that exhibit nearly equal peak to peak transmission enor. However, since spur gears were considered, the gear teeth were only profile modified. This does not imply that lead modifications are inappropriate for spur gears. In spur gears, lead modifications do not improve transmission enor, but spur gears may be lead modified to avoid adverse effects of shaft misalignment.

Fig. 2.3 shows contour plots of equal peak to peak transmission enor (PPTE) for a helical gear pair (geometry given in Table 2.1). The starting roll angle of modification was varied between the root and the tip of the tooth for both the pinion and the gear. The amount of relief at the tip of the pinion and the gear was varied between 0.0 and 0.0015". A parabolic lead modification of 0.0001" was applied at the end faces of gear tooth with 23 zero at the center of the face width. The lowest contour (PPTE = 6 microinches) encloses the optimum profile modification that minimizes transmission enor.

SAP Start Roll Angle of Modification TIP

Fig. 2.3 Contour plots of equal PPTE in microinches for various parabolic profile modification with a fixed lead modification of 0.0001"

In order to determine how profile and lead modification interact, the amount of total modification (lead + profile) at the corner where gear teeth come into and leave contact is fixed at 0.0014". The starting roll angle was varied between the root and the tip of the gear tooth. As the amount of lead modification is increased, the amount of tip relief in the profile modification is decreased such that total modification at the corner of the gear , tooth is kept constant. From Fig. 2.4, one can see that there exist many combinations of lead and profile modifications that exhibit nearly equal PPTE. We also find a need for a 24

large amount of profile relief and a small amount of lead crowning for reducing the transmission enor. It is seen that profile or lead modification alone would not be sufficient to achieve minimum transmission enor in helical gears.

0.0014" 0.0

c 0 •H 4-1 u(d 0) M-) •H vc 0. £ •H T) E-" (d 14-1 (D O J 4-> IW c O Q u c Q

0.0 0.0014" SAP Start Roll Angle of Modification TIP Fig.2.4 Contour plots of equal PPTE in microinches for various parabolic profile modifications with a fixed corner modification of 0.0001"

Fig. 2.5 shows the normalized load distribution over the zone of contact for the optimum profile modification in Fig. 2.3. The maximum load values (conesponding to 9) occur at the center of the zone of contact. Also, small load values are seen along the periphery of the zone of contact. More precisely, the small values along the periphery 25

show that the gear teeth were modified to minimize comer contact. The diagonal lines indicate loads for one out of 10 sets of contact lines in one mesh cycle.

P inion Tip 1 ~r e 1 I 1 1 1 1 0 2 2 1 4 3 3 4 2 1 3 3 4 4 4 A 4 4 t o 6 5 <> 6 6 3 8 4J 4 1 ? 8 7 5 o < 8 9 8 9 8 8 7 14-1 o <) 9 ? « o 8 8 Q) Contact Line 8 ? 7 y V ? e •aH J T 9 9 9 V 8 7 U-4 7 O B 8 © a 9 8 7 6 4-) 5 k 7 7 7 r 01 6 5 3 5 i t> 6 6 Q) 5 4 J 2 4 A •1 4 ,4 3 3 2 3 3 1 2 2 A 2 ' 0 9 1 1 1 1 1 Gear Tip

Face Width

Fig. 2.5 Normalized load distribution in the contact zone for optimum parabolic profile modification with a fixed lead modification of 0.0001"

Fig. 2.6 shows a plot of the optimum parabolic profile modification (optimum in Fig. 2.3) with roll angle on one axis and amount of relief on the other axis. It is very interesting to note that one could have achieved the same effect by applying relief at the tip and the root of the gear teeth. However, from a manufacturing point of view, it is usually easier to apply tip modification rather than root modification in gear teeth. 26

Gear Roll Angle in degrees

in CO 00 o vo m cs

Fig. 2.6 Optimum profile modification

Table 2.1 Gear Mesh Parameters

MESH PARAMETERS Pinion Gear Transmitted torque (in-lbs) 1500.0 Center Distance (inch) 2.926 Normal diametral pitch 1/in 10.0 Normal pressure angle (deg) 20.0 Helix angle (deg) 20.0 Profile contact ratio 1.49 Face contact ratio 0.82 Total contact ratio 2.31 Number of teeth 20 35 Face Width in inches 0.75 0.75 Outer diameter (inch) 2.328 3.925 Root diameter (inch) 1.876 3.472 Roll angle @ pitch circle 22.19° 22.19° Roll angle @ tipcircl e 35.14° 30.140 Roll angle @ SAP 8.27° 14.79° 27

0.0015"

0> •ri iH a •H EH IM 0 4->

ABOVE 2.7 0.0 SAP Start Roll Angle of Modification TIP

Fig. 2. 7 Load distribution factors for various parabolic profile modifications with a fixed lead modification of 0.0001"

Fig. 2.7 shows the contour plots of load distribution factor for various profile modifications. The load distribution factor is defined as the ratio of the maximum load intensity (load per inch of contact line) to the average load intensity over the zone of contact. The average load intensity is defined as total normal load divided by minimum theoretical length of line of contact (Lmin)- Fig. 2.7 shows that the load distribution factor is a minimum when the gears are relieved by a small amount to avoid load near the tip of the gear tooth. Interestingly, this is not the best case from a transmission enor (noise) point of view. 28

2.3 Cross modification Umezawa, et al. (1985) combined the lead and profile modifications in their tooth- flank modification method. Fig. 2.8 shows a typical modification of this type where modification direction is perpendicular to the line of contact in the contact zone. In the Load Distribution Program, this tooth flank modification (called cross modification) is characterized by two parameters, namely 1) Starting roll angle of modification and 2) the amount of comer relief.

Amount of

Fig. 2.8 Cross modification

One needs only two parameters instead of three to define a surface modification because modification direction is perpendicular to the contact line. The angle between the contact line and axis of gear rotation is equal to the base helix angle. The modification surface could be either linear or parabolic. This type of modification ensures that points on a contact line will have the same amount of modification. For a right hand helical gear pair, one relieves the left comer of the pinion tip and right comer of the gear tooth to minimize transmission enor. 29

0.002"

0) •H iH « U ad) u uo «4-l o 4-> a 0

0.0 SAP Start Roll Angle of Modification TIP

Fig. 2.9 Contour plots of equal PPTE in microinches for various parabolic cross modifications with a fixed lead modification of 0.0001"

Fig. 2.9 shows the contour plots of the PPTE when cross modification is used. A symmetric parabolic lead crown of 0.0001" is added to the cross modification. The starting roll angle of modification along the left face of the pinion tooth was varied from SAP to the tip of the gear tooth. The starting roll angle along the right face of the gear tooth was varied proportionately. The amount of comer relief was varied from 0.0 to 0.002". From Fig 2.9, we find that the optimum values are very sensitive to both variations in roll angle and in the amount of comer relief. This is evident from the high density of contour lines near the optimum. 30

0.002"

M-l 01 •H <-\ 0>

u o> c u o o

4-) « 3 Q

0.0 SAP Start Roll Angle of Modification TIP

Fig. 2.10 Load distribution factors for various parabolic cross modifications with a fixed lead modification of 0.0001"

Fig. 2.10 shows the contour plots of load distribution factor when the cross modification is used in conjunction with the lead modification. We do not find a large variation in the load distribution factor (around 1.3 in Fig. 2.10) for the region where transmission error is minimum (in Fig. 2.9). Fig. 2.11 shows the normalized load distribution when the optimum cross modification is applied. We only find small or zero load values at the corner and not around the periphery of the contact zone as observed for conventional modifications. This greater dispersion of load is one of the reasons why the load distribution factor has a low value for this type of modification. 31

Low load distribution factors are required for high bending strength and life. Hence one could apply cross modification when the aim is to minimize transmission enor without losing much in load sharing. However, applying cross modification would make the design very sensitive to manufacturing enors.

Pinion Tip

Gear Tip

Face Width *•

Fig. 2.11 Normalized load distribution in the contact zone for optimum parabolic cross modification with a fixed lead modification of 0.0001"

2.4 Topographical modifications This section deals with the evaluation of point-by-point tooth modification that gives zero PPTE. The procedure assumes a center-weighted load distribution in the zone of contact. For the assumed load distribution, the procedure evaluates the tooth deflections at discrete contact points along the lines of contact. The contact point where the maximum deflection occurs for a complete mesh cycle is defined as the reference point. 32

The initial separation (tooth modification) at the reference point is assumed to be zero. The tooth modification at the other contact points is equal to the difference in the deflections between the contact point and the reference point.

Pinion tip H 700-800

EH 600-700 c o • 500-600

L~3 400-500

D 300-400

CD c • 200-300

M 100-200

Gear tip • 0-100 Left Face width Right

Fig. 2.12 Contour plot of equal tooth modification in microinches over the contact zone

Fig. 2.12 shows the contour plot of tooth modification. Tooth modification at each contact point was evaluated and the tooth modification surface was curve fitted with a two dimensional quadratic regression analysis. Hence the modification shown in figure 2.12 would only exhibit near-zero PPTE. The gear tooth corners where mating teeth come into contact and leave contact require more modification than the other two comers. The cross modification in conjunction with lead modification comes close to representing 33 this type of modification. Manufacturing this type of 3-dimensional modification would be very difficult and would require a highly accurate point-by-point grinding machine.

2.5 Effect of Misalignment and Torque Variations This section deals with the effect of misalignment and torque variations on PPTE for gear meshes with optimum profile and cross modifications. Fig. 2.13 shows contour plots of equal PPTE for gear mesh with optimum profile modification for various values of misalignment and transmitted torque. Fig. 2.14 shows contour plots of equal PPTE for the gear mesh with optimum cross modification for various values of misalignment and transmitted torque. From Figs. 2.13 and 2.14, the cross modification is very sensitive to torque variations.

2500 ABOVE 46

(A ,Q r-l I a •H a •H o> & u 0 E-< -o o> 4J 4J •H g a(A HJ u EH

ABOVE 46 500 -0.0005 Misalignment Slope +0.0005 Fig 2.13 Contours of equal PPTE in micro inches for a gear mesh with optimum profile modification for various values of misalignment and torque. 34

2500

w Xi ABOVE 48 rH I

•H

Tl 0) 4-) 4J •H e CO c ABOVE 48 u EH

500 -0.0005 Misalignment Slope +0.0005 Fig 2.14 Contours of equal PPTE in micro inches for a gear mesh with optimum cross modification for various values of misalignment and torque.

2.6 Summary This chapter addressed the merits and demerits of three types of tooth modifications. One can minimize transmission enor effectively using any one of these three tooth modifications. Some of the conclusions are as follows: 1) One should apply cross modification when the aim is to minimize transmission enor without losing much in load sharing. Designs with cross modification are sensitive to manufacturing enors and torque variations.

2) When the aim is to minimize transmission error and its sensitivity to manufacturing enors and torque variations, conventional profile and lead modifications are recommended. 35

3) Before studying the sensitivity of three dimensional modifications towards manufacturing enors, one needs to investigate the effect of shaft misalignment and torque variations, and study its manufacturability issues. CHAPTER HI

UNCONSTRAINED STATISTICAL OPTIMIZATION

3.1 Introduction This chapter deals with minimizing the expected value of a function when the design space is not subjected to any constraints. Chapter 1 described the concepts of statistical optimization and presented a brief literature review of the methods used in statistical optimization. Methods described in chapter 1 were suitable for problems with few variables. This chapter presents a method that can be used to solve several variables. The methods dealt in this chapter are based on statistical principles used in design of experiments. Section 3.2 presents a generalized unconstrained statistical optimization problem with n variables and section 3.3 gives a brief background on design of experiments and explains the developed methodology.

3.2 Generalized Problem In statistical optimization, the goal is to minimize the expected value of a function.

Let f(xvx2,..jcn) be a function of n variables xltx2,..JCn that needs to be optimized. Let p(xl,x1,..jcn) be a joint probability density function associated with n random variables.

The expected value of a function /(jt,,jt2,..jc„) is given by

36 37

xl x2 X £[/(*! »*2»—*n)]= J J - •JJ/(*l>*2>-*» ) p(jCi,JC2,..Jf«)<&i

where Jt; and jt/ represent the lower and upper bounds for the ith random variable. This dissetation assumes that the n random variables are independent. Then using the statistical identity in equation 3.2

p(xl,x2,..JCn)=pl(xl)p2(x2)...pn(xn) (3.2)

equation 3.1 becomes

v+ v+ v+ *1 x2 xn E[f(xx,x2,..JCn)]= J j ... \f(xlyx2,..xn) p1(xl)p2(x2)...pn(xn)dx1dx2...dxn (3.3) xl x2 xn

In discrete form, equation 3.3 can be written as

x\ x2 xn Elfix^x^.JCn)] = X2-£/"(*i'*2.-*i!) Pi(xi)p2{x2)...pn(xn) (3.4)

Xi x2 xn Equation 3.4 gives the weighting factor used based on probability for each performance value f(xx,x2,..xn) to calculate the expected value of a function.

In most engineering problems, the probability distribution function for each variable is unknown. In cases where function /(jCpjCj,...*,,) is evaluated using computer programs, using equation 3.4 would be computationally time consuming. Methods like Monte-Carlo techniques could be used successfully to evaluate the equation 3.4. But, 38 such techniques would take a long time to yield results when simulation programs like LDP, finite element models are used to evaluate objective function. Section 3.3 presents a method based on statistical design of experiments to approximate equation 3.4 by an expression that will be easier to evaluate using computer programs.

3.3 Introduction to Design of Experiments Design of experiments is a statistical technique by which a designer acquires maximum information with a minimum number of experiments or trials. This approach evaluates the effects of many design variables simultaneously and has many advantages over methods that study one variable at a time. The method of studying one variable at a time seems to be accurate as one can get the performance as a function of the studied variable. However, such an approach inherently assumes that the effects of interactions between design variables is negligible. In addition, changing one factor at a time requires a large number of trials.

Let us consider three factors (variables), namely A, B and C. A linear model for the performance f is given by

f(Ai,Bj,Ck) = m + ai + bj + ck + (ab)ij + (bc)jk + (ac)ik + (abc)ijk + e. (3.5)

In equation (3.5), m indicates the overall average effect of all factors. In effect, m will be equal to the mean of the performance for all trials. The term ai denotes the deviation from mean m when factor A is set at level Ai. Similarly, bj, and Ck denote the deviations from mean m caused by setting factors B and C at level Bj and Ck respectively. The term (ab)jj denotes the deviation from mean m when factors A and B are simultaneously set at levels Ai and Bj respectively. This term along with the terms 39 (bc)jk and (ac)ik represent the 2-factor interactions and the term (abc)yk represent the 3- factor interaction. The term e stands for the enor in the repeatability of measuring function (f) for a given experiment.

A three factor analysis involves the study of the effect of 8 entities (excluding the enor). To study the effect of all 8 entities, one needs atieast 8 trials. An experiment that studies all entities is termed as a full factorial experiment. This concept can be explained graphically. For example, let the three factors have two levels each, namely a low level(- 1) and a high level(+l). Each trial can be represented by a point in a three dimensional space whose co-ordinates are either +1 or -1 in the each of the axis. A full factorial experiment in three-factor space provides the eight vertices of a cube. Figure 3.1 (a) shows a 23 experiment. When we have n factors, number of trials increases exponentially (2n trials). *£v1

3.1 (a). 23 experiment 3.1 (b). 22 experiment Fig. 3.1 (a & b)Full and fractional factorial experiments

Usually, engineering problems involve the main effects and the 2-factor interactions. The effects of higher order interactions are assumed to be negligible. In such cases, we do not cany out 2n trials , but 2n" trials (Box and Hunter, 1961 a & b). Such a design is termed as fractional factorial. Figure 3.1(b) shows a 22 experiment. One of the 40 disadvantages of using fractional factorials is that we will not be able to study the individual effects of factors. This is because we have 3 degrees of freedom (4 data points) to study the effect of 8 entities. The effects of main factors are confused(or confounded) with higher order interactions. Nevertheless, one assumes higher order interactions to have negligible effect.

Matrix experiment refers to a set of experiments(trials) where we change the values of design variables from one experiment to another. The aim in designing the matrix experiment will be to change the levels of factors in such a way, that atieast, the main effects (major factors of interest) are not confounded and can be estimated independently. In designs, where one can not avoid confounding, the interactions are assumed to have negligible effect.

Taguchi's Orthogonal Anay (OA) is a special case of these fractional factorial matrix experiments. As the name suggests, every column in an orthogonal anay is orthogonal to each other. Phadke (1989) explains orthogonality in combinatoric sense as for any pair of columns, all combinations of factor levels occur, and occur an equal number of times. This balancing property is a sufficient condition but not a necessary one. Even if the number of times a combination of factors occur is proportional throughout the design, one can call the design orthogonal.

Table 3.1 shows an L8 orthogonal anay. The left most column shows the trial number. The ones and twos in columns 1 to 7 represent factors at levels 1 and 2 respectively. Comparing columns 1 and 2, we find that "1" in the first column occurs with "1" in the second column twice. "1" in the first column occurs with "2" in the second column twice. This balancing property guarantees orthogonality. 41

For problems with 4 design factors (A, B, C and D), Taguchi recommends a L8 anay with factors A, B, C and D assigned to columns 1, 2 4 and 7 respectively. Such an assignment results in a design where the main effects are confounded only with 3-factor interactions. Assuming 3-factor interactions to have negligible effect, main effects can be studied independently. Interactions AB and CD are confounded and can be studied together using column 3. Similarly, interactions AC and BD is studied using column 5 and interactions AD and BC is studied using column 6.

Table 3.1 L8 Orthogonal Anay

Columns Trial No 1 2 3 4 5 6 7 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 3 1 2 2 1 1 2 2 4 1 2 2 2 2 1 1 5 2 1 2 1 2 1 2 6 2 1 2 2 1 2 1 7 2 2 1 1 2 2 1 8 2 2 1 2 1 1 2

The concept of a matrix experiment discussed so far is used to evaluate the effects of each factor on performance f(xl,x2,..jcn). Based on the analysis, one can determine those factors that influence the performance most. Based on the results, one chooses levels(or values) for the design variables(or process parameters) to optimize performance. For example, if performance f is the largest at level Ai, and the objective is to minimize performance f, then we would choose A2 for factor A. When factors are modeled with 42 two levels, one can study the linear effects of factors, i.e., performance is either monotonically increasing or decreasing with the levels of that factor. By choosing three levels for each factor, we can study the non-linear effects of each individual factors. Taguchi has proposed 3-level anays that can be used to study the non-linear effects of design variables.

In summary, this section dealt with how one can estimate the effects of design variables. In many cases, one should model the uncertainties(variance or "noise") in each design variables to study the effect of these noises on performance f. The next section deals with the use of concepts presented in this section to design devices such that performance is least sensitive to uncertainties/enors in design variables.

3.4 Procedure The aim of this section is to approximate equation (3.4) by an expression that will be easier to evaluate when using performance simulation programs. Sundaresan et al. (1989) evaluated the function at all the worst combinations of design variables and defined a root mean square value of the difference between the function at the worst cases and the function value at the target as a measure of the sensitivity. Equation 3.6 gives the sensitivity index (SI) as the measure of sensitivity. (fi fc)2 Hil ~ (3.6)

The objective function (F) during statistical optimization is a linear weighted function of the target performance( fc) and the sensitivity index. Equation 3.7 gives the objective function. 43

F = afc +(l-a)SI 0£a

As stated in chapter 1, in this method, the number of function evaluations increase exponentially as the number of variables increase. This deficiency can be overcome by applying concepts described in section 3.3 in defining the sensitivity index of a design. Since equation 3.6 considers all worst combinations of design variables, we refer to it as a full factorial design. The use of fractional factorials instead of full factorials reduces the number of function evaluations in computing the sensitivity index. The orthogonal anays proposed by Taguchi help us in designing our fractional factorials. However, as stated earlier, fractional factorials assume negligible interactions between some design parameters. L4 orthogonal anays require 4 function evaluations and accommodate up to 3 design parameters. Equation 3.6 conesponds to a L4 anay applied to 2 design parameters. Using L4 anay for three parameters saves 4 function evaluations when compared to the full factorial design.

For problems with more than 3 but less than 8 design parameters, L8 orthogonal anay is suitable. When there are 4 design parameters, one can assign the parameters to columns 1, 2,4 and 7 so that the main effects (effect of individual design parameters) are only confounded with the three-parameter interactions. Assuming the three-parameter interactions have negligible effect, the effect of each parameter on SI can be computed.

After evaluating the function at n worst combinations of design variables, equation

(3.8) yields the sensitivity index. Equation 3.6 defines SI using fi and fc values while equation 3.8 uses only fj values. Since the value of SI is independent of the value fc, the weight factor a can take values between 0 and 1. The function F defined by equation 3.7 serves as the objective function during optimization. 44

s/ = (3.8)

If the aim is to penalize larger values of fi, one can define SI as

SI = (3.9) . n.. \ i=l )

This dissertation adapted the Broydon-Fletcher-Goldfarb-Shanno variable metric method (Vanderplaats, 1984) during computation to determine the statistical optimum.

3.5 Helical Gear Example This section presents an helical gear example to illustrate the method discussed in this chapter. Table 3.2 gives the geometry of the gear mesh. We considered four variations, namely, 1) Variation of 0.00015 in parabolic tip relief 2)Variation of 1.5 degrees in starting roll angle , 3) Torque variation of 200 in-lbs and 4) shaft misalignment of 0.0005 inch per inch of facewidth. During optimization, we varied the following five variables independently:

1) Starting roll angle of modification on the pinion from SAP to the tip of the tooth. 2) Starting roll angle of modification on the gear from SAP to the tip of the tooth. 3) Amount of tip relief on the pinion tooth from 0.0 to 0.0015". 4) Amount of tip relief on the gear tooth from 0.0 to 0.0017". 5) Amount of lead modification at the two end faces of the pinion tooth from 0.0 to 0.0005". The lead modification was parabolic with zero at the center of the face 45 width. The amount of modification at the two end faces of the pinion tooth was assumed to be equal and the gear tooth was unmodified in the lead direction.

Table 3.2 Gear Geometry

GEOMETRIC PARAMETERS Pinion Gear Transmitted torque (in-lbs) 750.0 Center Distance (inch) 2.7953 Normal diametral pitch (1/in) 12.05 Normal pressure angle (deg) 16.0 Helix angle (deg) 30.0 Profile contact ratio 2.02 Face contact ratio 1.13 Total contact ratio 3.15 Number of teeth 18 41 Face width 0.6 0.6 Outer diameter (inch) 2.066 4.005 Root diameter (inch) 1.5517 3.4911 Roll angle @ pitch circle 16.70° 16.70° Roll angle @ tip circle 44.09° 22.440 Roll angle @ SAP 3.63° 4.680

The weighting factor a is set at 0.5. Table 3.3 shows the results of optimization with the values of design parameters for the peak and statistical optimum. Figure 3.2 shows the optimization results graphically. For a small increase in PPTE, we find a considerable decrease in the sensitivity index. Figure 3.3 shows the worst possible performance for the peak and statistical optimum. In figure 3.3, if the target is the peak optimum, PPTE is as high as 42.0 micro inches for the worst design. However, if the target is the statistical optimum, worst value of PPTE is only 25.0 micro inches. 46

Table 3.3 Comparison of peak and Statistical optimum

PARAMETERS Peak Statistical Start roll angle - pinion (deg) 11.115 15.892 Start roll angle - gear (deg) 13.63 12.709 Amount of relief - pinion (inch) 0.001357 0.001128 Amount of relief - gear (inch) 0.001524 0.001029 Lead modification amount (inch) 0.000347 0.00014 PPTE (u ins) 6.65 10.47 Sensitivity Index (|i ins) 26.542 15.922 Function F (|i ins) 16.596 13.196

HPEAK OPTIMUM B STATISTICAL OPTIMUM

PPTE

Figure 3.2 Optimization results

45-r

• Worst of L8 • Target

Peak Statis tical

Figure 3.3. Worst case of L8 for peak and statistical optimum 47

3.6 Discussion of Results This section performs an analysis of variance (ANOVA) for the peak and the statistical optimum to determine which variation affects PPTE the most. ANOVA is a technique that breaks the total variation into accountable variations. The optimization process used an L8 orthogonal anay to evaluate the sensitivity index. The use of an L8 orthogonal anay required the evaluation of PPTE at only 8 out of 16 possible worst combinations of design variables. To be sure that remaining unchecked 8 worst cases do not show performances poorer than the peak optimum, PPTE was evaluated at all the 16 possible worst combinations for the peak and statistical optimum. Table 3.4 compares the PPTE at all the 16 worst cases for the peak and the statistical optimum.

Table 3.5 gives the variance caused by each parametric variation. Appendix A gives a brief introduction to analysis of variance and presents all the equations used to compute the data shown in Table 3.5. The total variance (sum of squares) for all factors together for peak optimum is 1440 where as for the statistical optimum is only around 418. This shows that the procedure has achieved its objective in minimizing the total variance during statistical optimization.

Let us now look at the effects of each factor. For the peak optimum, variations in tip relief and torque have the most significant effects and contribute to nearly nearly 93% of total variance in performance. The interaction between tip relief and torque (Relief_X_Torque) has a larger effect than the main factors (tip relief and torque). The effect of interactions between design variables contributed to a significant improvement T during statistical optimization. 48 Table 3.4 PPTE at all worst combinations for peak and statistical optima

Trial Roll Angle Relief Torque Alignment Peak Statistical 1 + + + + 18.70 16.99 2 + + + - 18.89 15.40 3 + + - + 41.51 10.38 4 + + - - 39.17 11.07 5 + - + + 20.78 18.95 6 + - + - 25.10 25.52 7 + - - + 19.99 12.10 8 + - - - 19.25 9.35 9 - + + + 12.79 18.87 10 - + + - 16.49 17.77 11 - + - + 41.75 13.92 12 - + - - 38.98 11.10 13 - - + + 15.49 18.46 14 - - + - 25.07 23.3 15 - - - + 21.72 12.3 16 - - - - 20.94 5.83

Table 3.5 Sums of Squares for peak and statistical optima

SUM OF SQUARES FACTOR PEAK STATISTICAL All Factors 1440.85 418.67 Roll Angle 6.45 0.20 Tip Relief 224.55 6.64 Torque 506.25 299.38 Alignment 7.78 0.43 Relief_X_Torque 613.55 36.27 49 For the statistical optimum, 75% of the total performance variance is caused by variation in torque. Variation in tip relief does not affect the performance variation significantly for the statistical optimum. The amount of tip relief for the statistical optimum is lesser than that of the peak optimum. The reduction in the amount of tip relief lead to the decrease in its effect on total performance variation. Hence, one should choose "lower than optimal" modification over "larger than optimal" modification when addressing the effect of variations in tip relief on PPTE. However, care has been taken to minimize comer contact. The amount of tooth deflection is around 0.0008" and sum total of relief due to profile and lead modifications at the comer of the gear tooth where contact initiates and ends is greater than 0.0008" and thus comer contact is minimized.

Note that the misalignment has very little effect on total performance variation for the peak and the statistical optimum. This could be due to the fact that tooth deflections are much larger than the value for misalignment used during analysis. The peak optimum has a well modified gear tooth in the lead direction and hence misalignment has very little effect.

3.7 Summary This chapter presented a technique for unconstrained statistical optimization and illustrated it using an helical gear example. The technique used fractional factorials and only looked a fraction of the worst combinations of design variables during optimization. This reduced the number of function evaluations significantly. This chapter illustrated the procedure using a helical gear example. The statistical optimum showed a considerable decrease in performance variance and performance at the worst combination of design variables over the peak optimum. The chapter used analysis of variance to 50 identify the variance caused by the variations in each design variable. The next chapter will extend the concepts discussed in this chapter to constrained optimization. Chapter IV

CONSTRAINED STATISTICAL OPTIMIZATION

4.1 Introduction This chapter presents a technique to solve a non-linear statistical optimization problem involving variations in design variables and constraints. Chapter 3 dealt with unconstrained optimization where the aim was to minimize the expected value of a function (PPTE) for a bounded but unconstrained design space. This chapter extends the concepts described in chapter 3 to constrained problems.

The rest of this chapter is organized as follows: Section 4.2 gives a brief background on constrained statistical optimization and defines the conditions for criticality of constraints. Section 4.3 discusses the work reported in the literature in computing the sensitivity of the optimal design to changes in problem parameters. Section 4.4 describes three techniques for solving constrained statistical optimization problems and discusses the merits and demerits of each of them. Section 4.5 illustrates the three methods presented in section 4.4 with a spur gear example.

51 52 4.2 Background A general non-linear optimization problem is stated as follows: Minimize F(x) subject to gj(x, bj) = gj(x) - bj < 0.0 j=l,2 m

hk(x, Ck) = hk(x,) - Ck = 0.0 k=l,2 1 x e S where S=|x:x€l\.n x^x^x"}

x=[x1,x2,...xn] 1 T u u T x =[xl,xi,...x!1] x =[Xl ,x^...xS] (4.1)

Similarly, a general non-linear statistical optimization problem may be stated as Minimize E[F(x)] subject to E[gj(x,bj)]<0.0 j=l,2 m E[hk(x,bk)]=0.0 k=l,2 1 X € S where S=|x:xeR.n x^x^x"}

x=[x1,x2,...xn]

x =[^x1,x2,...xnJ x =|^x1,x2,... xnJ

with variations Axi and Abj in XJ, (i=l, 2,..n) and bj respectively. (4.2)

The technique presented in chapter 3 evaluated the expected value of the objective function by applying statistical design of experiments concepts. One could also use the 53 same concepts to evaluate the expected value of a constraint too. However, the use of expected values for a design constraint implies that the constraints are satisfied in a probabilistic sense. In a strict sense, the constraints are violated for some worst combinations of the design variable. For an inequality constraint, E[gj(x, bj)] < 0.0 would mean that approximately 50 % of the time, the design constraint would be violated. Instead, one prefers the worst combinations of design variables to satisfy the design constraints. Hence there is a need to define when a constraint is active or violated for statistical optimization. This leads us to the following definitions of design spaces and criticality of a constraint:

Definition 1: Tolerance Space and Corner Space Tolerance space (T) is a set of points close to the target design point where each point represents a possible combination of design variables due to uncertainties in each design variable. The shaded rectangle in Fig. 4.1 represents the tolerance space for 2-D problems. The target design point is represented by a small rectangle in the center of the shaded rectangle.

Each target design point xt = [xlt,x2t,... xnt] € S has a tolerance space

associated with it.

T(xt)=|x:xeS| |x-xt|

where Ax = [Ax1,Ax2,... Axn] (4.3)

The comer space (W) consists only of comer vertices of the tolerance space. In Fig. 4.1, the 4 comer vertices of the shaded rectangle are the elements of the comer space. 54

W(xt)={x:x e S | |x-xt| =Ax } (4.4)

Theorem 4.1

If a constraint is monotonic with respect to all design variables in the tolerance space, then the maximum constraint value will occur at one of the corner points.

Proof: Consider a vector x of two design variables (xi, X2) with target values x 5= x x t ( lt» 2t)- Assume that constraint g = g(x1,x2) is monotonically increasing in x\ and monotonically decreasing in X2.

Since g is monotonically increasing in xi and monotonically decreasing in X2, the following equations are satisfied:

g(xj",x2) > g(x1,x2) V(Xl,x2)€T(xlt,x2t) (4.5)

g(Xl,x2) > g(x1,x2) V(Xl,x2)eT(xlt,x2t) (4.6)

Substituting for x2 = x2 in eq(4.5) and using eq(4.6) we get

g(xi\x2) > g(x!,x2) > g(x!,x2) V(x1,x2)eT(xlt,x2t) (4.7)

which proves that the maximum occurs at one of the vertices. The proof of this theorem can be extended to n design variables easily. 55 Definition 2: Statistically Active Constraint An ith inequality constraint is considered active when the value of the constraint is zero at some of the worst combinations of design variables and negative at the target design and at the remaining worst combinations of design variables.

For xt e S, g, is statistically active if

1) gi(it)< 0.0 z, w (4.8) 2) 3xeW(xt) s.t. gj(x) = 0.0 3) Vxe T(xt) gi(x)<0.0

Definition 3: Quasi-Active Constraint: An itn inequality constraint is considered quasi-active when value of the constraint is less than zero at the target design point and positive for some worst combinations of design variables.

For xt e S, gj is quasi - active if

1) gi(xt)<0.0 (4.9)

2) 3 xeT(xt) s.t. gi(x)>0.0

Definition 4: Peak Active constraint An i^ inequality constraint is considered peak-active when the value of a constraint is greater than zero for some of the worst combinations of design variables and zero for the target design point. This would be an outcome in an optimization process if we did not model variations in constraints due to the deviations in design variables. 56 For x e S, gi is peak - active if t (4.10) 1) gl(xt) = 0.0

constraint constraint constraint

feasible feasible region region

constraint constraint constraint Statistically Active Quasi Active Peak Active Constraints Constraints Constraints

constraint constraint

constraint constraint Quasi Violated Violated Constraints Constraints

Fig. 4.1 Criticality of constraints

Definition 5: Quasi- Violated Constraint An itn inequality constraint is considered quasi-violated when the target design point violates the constraint and some of the worst combinations of design variables do not. 57

For xt € S, gj is quasi - violated if

1) gi(xt)>0.0 (4.11)

2) 3xeT(xt) s.t. gi(x)£0.0

Definition 6: Violated Constraint An im inequality constraint is considered violated when all the points in the tolerance space violate the constraint.

For xf € S, g; is violated if W T (4'12) gi(x)>0.0 VxeT(xt)

Fig. 4.1 shows the various cases of criticality for inequality constraints. For an equality constraint, variation in the design variable would cause a violation of the constraint. Hence we do not describe any special definitions for activeness of an equality constraint. For an equality constraint, the procedure will assure that the target design point satisfies the equality constraint.

4.3 Literature Review As stated in chapter 1, one of the earliest forms of sensitivity analysis was performed using Lagrange multipliers. Lagrange multipliers give a unit change in the optimal function for an unit change in the constraints. Fiacco (1968) and Sobieski et al. (1982) provide methods for evaluating sensitivities of the optimal objective function and optimal design to changes in design variables and parameters that were kept constant during optimization. These methods were based on differentiating Karush-Kuhn-Tucker (KKT) conditions of optimality with respect to the fixed design parameters and design variables. 58 Vanderplaats (1985) also provided a method to estimate the sensitivity of the optimum design to changes in fixed parameters based on the method of feasible directions. In this method, the fixed parameter is modeled as an additional design variable and hence this method is also called Extended Design Space (EDS) method. One advantage of this method over methods based on KKT optimality conditions is that this method does not require second derivatives of the objective function and constraints.

Beltracchi and Gabriele (1988a and b) presented a method based on Recursive Quadratic Programming (RQP) method to estimate sensitivity derivatives without having to evaluate the second derivatives of the objective function. In their method, for each parameter (p), they performed two optimizations using RQP method at the upper (p+) and the lower (p~) variations of the design variables to evaluate the optimum functions f+ and f" respectively and evaluated the sensitivity derivative using a central difference scheme.

The above methods can be used successfully to evaluate sensitivity derivatives in a non-linear optimization problem after optimization is completed. Beltracchi and Gabriele (1988b) briefly discussed the merits and demerits of each of these methods. However, in statistical optimization, we are interested not only in sensitivity derivatives but also in finding the feasible combination of design variables that minimize objective function and its sensitivity to design parameters. There is a need to incorporate the calculation of sensitivity derivatives during the optimization procedure rather than after optimization.

To minimize a function with variations in constraints, Parkinson et. al. (1990) advocate a two step solution method. The first step addresses the optimization problem with only the nominal constraints and variation (Ab) and does not include the function variations. The optimum achieved at the first step is called nominal optimum. The 59 function variations are evaluated at the nominal optimum. The statistical optimum is assumed to be close to the nominal optimum and hence they assumed that the constraint variations to be a constant in the domain containing both the peak and statistical optimum. The second step involves optimizing the problem with constant constraint variation built into the model. A major limitation in this two-step procedure is that one can not study the effect of variations in individual variables on constraints.

In statistical optimization, one should be interested not only in minimizing the effect of variations (tolerances) in design variables on performance, but also in controlling the effect of variations in design variables on the active constraints. The next section presents three techniques for statistical optimization to compute a feasible, robust and optimal combination of design variables.

4.4 Procedure 4.4.1 Method 1: Using heuristics This method is a single step optimization procedure used to achieve feasible robust optimum. This method evaluates the objective function and constraints at the worst combinations of design variables. A constraint (gj(x) - bj < 0.0) during peak optimization would be modeled during statistical optimization as

Max{[gj(y)-bj],Vy€W(x)} < 0.0 (4.13)

The expected value of objective function would be evaluated using Taguchi's Orthogonal anays as discussed in chapter 3. The problems solved during statistical optimization can be stated as Minimize E [ F(x) ] 60 Subject to

Max{[gj(y)-bj],VyeW(x)} < 0.0

X € S where S=|x:x€Kn x'^x^x"!

x=[x1,x2,...xn] 1 T u n T x =[xi,xi....xJ1] x =[x1 >x5,...xS] (4.14)

with variations AXJ and Abj in XJ, (i=l, 2,..n) and bj respectively.

This method, equivalent to brute force technique, is very simple to implement in a computer program because it does not require second derivatives of either the objective function or the constraints. One of the drawbacks of this method is the need to evaluate the constraints at all the worst combinations of design variables. This method is suitable for problems where evaluation of the constraints is not computationally time consuming. For problems with n variables, one would have to evaluate constraints at 2n worst combinations of design variables.

In order to reduce the number of constraint evaluations during optimization for problems with a large number of design variables, one could use Taguchi's orthogonal anays and evaluate constraints at only a fraction of the worst combinations of the design variables. After evaluating the constraints at the selected worst cases, we use statistical technique (Phadke, 1991) to predict which other worst cases would show larger constraint values and evaluate the constraint only at those predicted worst cases. Theorem 4.1 shows that if one assumes monotonicity in the tolerance space, one of the worst combinations of design variables will exhibit the maximum constraint value. This technique is similar but superior to evaluating gradients of constraints using finite 61 difference methods because the finite difference technique changes one variable at a time and thus lacks information on interactions between design variables.

In our applications, the evaluation of constraints (bending and contact stresses) take about less than 0.1% of time it takes to evaluate a performance function. The solution of the optimization problem usually has 2 to 3 active constraints. Method 1 is suitable for such applications.

4.4.2 Method 2: Constraints with built-in constraint variation This method is an off-shoot of the brute force method. Instead of using orthogonal anays to determine which worst combination of design variables has a larger constraint value, we use the gradient information to evaluate the worst possible value of the constraint. The maximum possible variation in constraint g due to deviation A\\ in design variables XJ (i=l, 2, ...n) is approximated using equation (4.15).

agj AX: (4.15) i=i dxj

In this method, we add this variation to the constraint formulation and solve the following optimization problem:

Minimize E[F(x)] Subject to n gj(X'bi) + I < 0.0 i=l OXj X € S n where S=|x: xeR Ix^x^x"! 62

X=[x1,X2,...xn] 1 B B T x =[xl,x2,...x!1f x =[x1 fxS,...xS] (4.16)

One could also replace the constraint in the above equation by

Equation (4.16) is more conservative than equation (4.17). Modeling constraints using equation (4.17) could result in quasi-active constraints.

4.4.3 Method 3: Approach using KKT conditions This method is based on differentiating KKT conditions of optimality with respect to design variables. In this method, the statistical optimization procedure consists of two steps. The first step involves finding the robust optimum without considering any variation in the constraints. However, due to inherent variations in the constraints, the optimum achieved at this stage is infeasible. The second step involves finding the change in the optimum design variables for a change in the constraints.

The first step solves a constrained optimization problem with objective function E[F(x)], where E[F(x)] is the expected value of a performance characteristic. The procedure uses Taguchi orthogonal anays to approximate the expected value of a performance characteristic. Algorithms like Augmented Lagrange Multiplier (ALM) methods along with BFGS for search directions, can be used to find the constrained optimum. Appendix B outlines ALM method and BFGS variable metric method. At the end of the first step, optimum design variables and language multipliers associated with 63 active constraints are available. However, if the method used for the first step does not yield Lagrange multipliers, one can estimate Language multiplier by

T T A = -[A A]~ A VXF (4.18)

Where A is a vector of Lagrange multipliers and matrix A contains the gradient information of the active constraints. Element Ay of matrix A is given by

Aij = |i (4.19)

The second step involves calculating the following partial derivatives:

^ and |^ j=l,2, m (4.20) dgj 3gj Since for active constraints gj = bj, derivatives in eq. (4.20) are equal to

?*- and ^- j=l,2 m (4.21) 3bj 8bj J

In equations (4.20) and (4.21), X* and x* are the Lagrange multiplier and the constrained optimum based on the Lagrangian given by equation (4.22).

in L(x,A,,b) =F(x) + 5>jSj(x'bj) (4.22)

To calculate the partial derivatives Vbx and VbA,* in equation 4.21, this method differentiates the KKT conditions with respect to bj. This ensures that KKT conditions 64 are satisfied for various values of bj. Equations (4.23) and (4.24) give the KKT conditions.

m VxL = VxF + 2XjVxgj(x,bj) = 0 (4.23)

Xjgj(x,bj) = 0.0 For j=l,2, m (4.24)

Differentiating Kuhn-Tucker conditions with respect to b, we get

" V xT " VXL Vxgf VxgJ ¥ x&m b VxbL T Wgi gi 0 0 Vb*i ^iVbg? V T (4.25) *<2 xg2 0 g2 0 Vb^2 = —

Am^xgm 0 gm . Vb^mT. >mVbgm.

Since Vx L is not an explicit function of bj, Vxb L is a zero matrix. Since only active constraints are considered, gj = gj(x) - bj =0.0

1 J V^L A Vbx T (4.26) A 0 vab?

where A is a mxn matrix with each Aij term given by equation (4.19), I is an identity matrix of size mxm and Hessian Vx L is given by

32L V^L= (4.27) SXJBXJ 65 After solving for Vpx and VbA,, one can evaluate the change in x* and X for changes in constraint g using.

m dX*:1 ™ 3x* A^j=-I^ Agk Ax^-X^-Ag, (4.28) k^^k k=ldDk where the variation in the constraint Agi is given by equation 4.15.

The new optimum is given by

*new = Xok, + Ax (4.29)

One of the advantages of this approach is that it reduces the number of constraint evaluations during optimization. This is because this approach does not consider the variation in the constraints in the first step. However, this technique fails if one of the inactive constraints is violated in the new optimum.

4.4.4 Discussion The first method based on brute force with heuristics and the second method based on first order approximation of the constraint variation are ideal for problems where evaluation of the constraints is easy and not time-consuming. The third method based on differentiating KKT conditions is suitable for problems where constraint evaluation is computationally expensive. However, the third method is a two step procedure where the method assumes that there is no change in the set of active constraints during the second step. This problem however does not occur in the first two methods. In cases where the evaluation of constraints involve analysis procedures (like FEM) that are time consuming, it is recommended that one should come up with simple algebraic equations 66 by using regression analysis (Bhakuni, 1991). Using curve-fitted equations significantly reduces computing time.

4.5 Spur Gear Example In this section, we illustrate the methods discussed in section 4.4 using a spur gear example. In this example, the aim is to design a spur gear mesh with minimum transmission enor and at the same time, is not sensitive to enors in pressure angle and uncertainty in number of teeth due to its discrete values. The problem specifications used are: 1) Transmitted torque of 1250 in-lbs 2) Center distance of 3.5 inches 3) Gear ratio of unity

During optimization, the following variables were varied:

1) Number of teeth (Np)was continuously varied between 20 and 80. 2) Generating pressure angle (<()„) was varied between 10 and 25 degrees.

The following parameters were assumed during optimization: 1) The face width equals 1.0 for both the pinion and the gear.

2) The tooth thicknesses of the pinion and the gear were reduced by 0.0025/Pnd at the standard pitch diameter to account for positive backlash. 3) This example considered only standard gears with the operating pitch diameter equal to the standard pitch diameter. The addendum coefficient of the pinion and the gear was 1.0. 4) This example used standard hobs with hob addendum coefficient of 1.25 and hob tip radius coefficient of 0.157. 67 5) The manufacturing variation in pressure angle is 0.3 degrees and uncertainty in the number of teeth is 0.5

An unique combination of number of teeth and pressure angle represents a target gear design. The procedure evaluated PPTE for the target gear mesh. The procedure computed sensitivity index using eq. (3.8) by evaluating PPTE at the 4 worst cases (due to variations in pressure angle and uncertainty in number of teeth) given by the L4 orthogonal anay. We assumed a to be 0.5 and used eq. (3.7) to evaluate the objective function.

Following were the constraints modeled during optimization: 1) Minimum number of teeth to avoid undercut. This is a function of pressure angle, hob shift to account for backlash, hob addendum and hob tip radius.

2) Minimum top land thickness for the pinion of 0.3/Pnd where Pnd is the normal generating diametral pitch.

3) Minimum top land thickness for the gear of 0.3/Pnd. 4) Bending stress of the pinion should be less than 30 Ksi. The procedure evaluated the bending stress using the AGMA geometry factor method (Erichello ,1981).

Fig. 4.2 shows contour plots of equal objective function values for various combinations of number of teeth and pressure angle. The two dark lines represent the bending stress constraint and the constraint that represents minimum number of teeth to avoid undercutting. The optimum point (P) lies in the intersection of these two constraints. The optimization process without modeling variations in constraint converged to point P (peak optimum). 68

10 Pressure Angle in degrees 25

Fig. 4.2 Constraints and contours of objective function

Table 4.1 shows the results of the methods described in section 4.4.

METHODS USED FUNCTION Nn n #FE #GE Peak Optimum (Point P) 53.416 58.21 11.36 188 188 Method 1 (Brute force) 53.735 57.84 11.75 193 772 Method 2 (using eq. 4.16) 53.726 57.82 11.74 193 579 Method 2 (using eq. 4.17) 53.662 58.00 11.68 196 579 Method 3 (Using KKT) 53.735 57.84 11.75 189 191

Table 4.1 tabulates the results of the three methods discussed in section 4.4 where #FE and #GE represent the number of function and constraint evaluations. In method 1, 69 we evaluated at all of the 4 (22) worst combinations of design variables and hence the statistical optimum achieved in method 1 is the optimum and constraints were statistically active. Two cases were run for method 2, one using eq. (4.16) and the other using (4.17). Method 2 using eq. (4.16) converged to an optimum close to the one achieved in method 1. The results achieved in method 2 using eq. (4.17) were in between the peak and the statistical optimum. The result achieved using method 3 is very close to the optimum in method 1. However, this is coincidental and one cannot conclude that method 3 will always yield better results than method 2. The number of function evaluations (# F.E) is nearly the same for all the methods but method 3 requires less number of constraint evaluations (# G.E).

This example treated the number of teeth to be a continous variable and the optimum value for number of teeth was 57.84. However, in practice, a designer will choose the number of teeth as 58. Since the uncertainity in the number of teeth chosen was 0.5, 58 will lie within the tolerance space and constraints will not be violated.

4.6 Summary In summary, chapter 4 introduced concepts in constrained statistical optimization and discussed three solution techniques. This chapter illustrated the techniques using a spur gear example with two design variables. The three solution techniques produced nearly identical results. Two major contributions of this research in this chapter are 1) Definition of the statistical optimization problem and 2) Use of the three solution techniques to solve constrained optimization problem. The next chapter will apply the techniques presented in this chapter and chapter 3 to study the effect of geometric design variables on transmission enor. CHAPTER V

PARAMETRIC STUDY USING HXUSTRATIVE EXAMPLES

5.1 Introduction This chapter applies the techniques presented in chapters 3 and 4 to obtain spur and helical gear designs having minimum transmission enor. The aim is to study the effect of design variables like number of teeth, pressure angle, diametral pitch, helix angle, hob shift and working depth on transmission enor through illustrative examples. Sections 5.2 deals with design of spur gears with minimum transmission enor using methods discussed in chapters 2 and 3. Sections 5.3 and 5.4 focus on designing spur and helical gears respectively with minimum transmission enor and balanced bending strength. Section 5.5 deals with the design of helical gears that not only have minimum transmission enor but also are less sensitive to manufacturing enors in pressure angle and helix angle. Section 5.6 briefly discusses the effect of diametral pitch and working depth on transmission enor in non-standard spur and helical gear designs.

5.2 Example #1 Chapters 3 and 4 dealt with minimizing transmission enor in gears but differed in their approaches. In Chapter 3, the procedure minimized PPTE by applying profile and

70 71 lead modifications while in chapter 4, the procedure minimized PPTE by finding the optimum combination of geometric design variables (number of teeth and pressure angle). This example shows that one can minimize transmission enor in spur gears in two steps and there is no need for combining the two optimization procedures into a single step procedure.

The procedure used in this example consists of three steps. The first step determines the optimum combination of number of teeth and pressure angle that minimizes transmission enor for gear meshes with no tooth modifications. The second step calculates the optimum profile and lead modification that minimizes transmission enor for the gear mesh defined by the optimum in the first step. The third step finds the optimal combination of number of teeth and pressure angle that minimizes transmission enor for gear meshes with optimum profile and lead modification found in the second step.

The specifications for this example are: 1) Transmitted torque of 1250 in-lbs 2) Center distance of 3.5 inches 3) Gear ratio of 1.0

The optimization procedure varied the following design variables:

1) Number of teeth on the pinion (Np) was continuously varied between 20 and 80. 2) Normal Generating pressure angle (<)>„) was continuously varied between 10 and 25 degrees.

This example assumes the following values for the following parameters: 72 1) The face width equals 1 inch for both the pinion and the gear.

2) The tooth thicknesses of the pinion and the gear are reduced by 0.0025/Pnd at the standard pitch diameter to account for a positive backlash. 3) This example considered only standard gears with the operating pitch diameter being the same as the standard pitch diameter. The addendum coefficient of the pinion and the gear is 1.0. 4) Standard hob cutters were used with hob addendum coefficient of 1.25 and hob tip radius coefficient of 0.157. 5) The manufacturing variation in tip relief and operating variation in torque are 0.0002" and 250 in-lbs respectively. Maximum possible shaft misalignment used is 0.0005 inch per inch of facewidth. The variation in the start roll angle of modification is zero. 6) This example applies parabolic tip modification with the starting radius of modification at the highest point of single/double tooth contact (HPSTC/HPDTC). The starting position of parabolic lead modification is at the center of the facewidth and the two ends of the facewidth had equal amounts of lead modification. 7) No variations in the constraints were modeled in this example

2) Minimum top land thickness for the pinion of 0.3/Pnd-

Fig. 5.1 shows contour plots of equal objective function values for various combinations of number of teeth and pressure angle. A unique combination of number of teeth and pressure angle represents a target design. Each target design has no tooth modifications. Equation (3.8) calculates sensitivity index for each target design by evaluating PPTE at the 8 worst cases (due to variations in profile, torque and shaft misalignment) given by the L8 orthogonal anay. Equation (3.7) evaluates objective function (F) using weighting factor a = 0.5. The two dark lines represent the bending stress constraint and the constraint that reflects the need for a minimum number of teeth to avoid undercutting. The optimum point (P) lies on the intersection of these two constraints.

10.0 Pressure Angle in degrees 25.0 Fig. 5.1 Constraints and contours of objective function 74 Fig. 5.2 shows the results of the second step. This step applies tooth modifications in the form of tip relief and lead modification for the gear mesh given by the optimum point P in Fig. 5.1. On one axis, the amount of tip relief was varied between 0.0 and 0.001" and on the other axis, the amount of lead modification was varied between 0.0 and 0.001". The lowest contour (value=19) encloses the optimum combination of profile and lead modification that minimizes the objective function. In Fig. 5.2, the optimum point is represented by point O. Note that the objective function is less sensitive to changes in the lead modification but is very sensitive to changes in the amount of tip relief.

0.0 Amount of Lead Modification 0.001" Fig. 5.2 Contours of equal objective function value for optimum design in Fig. 5.1 with various profile and lead modifications

Fig. 5.3 shows the results of the third step, which finds the optimal combination of number of teeth and pressure angle that minimizes transmission enor for the gear mesh with optimum profile and lead modification (point O in Fig. 5.2). Table 5.1 gives the 75 geometry data of gear mesh (represented by point P) and optimum tooth modification. As the gear and pinion are identical, Table 5.1 gives only values for the pinion. The optimum point in the Fig. 5.3 is the same as optimum point in Fig. 5.1. Hence one can conclude that we can design spur gears with minimum transmission enor by first determining the optimum combination of geometric design variables (number of teeth, etc.) and then applying optimal profile and lead modification. This infact validates the cunent practices followed in the gearing industry.

10.0 Pressure Angle in degrees 25.0 Fig. 5.3 Constraints and contours of objective function for optimum profile modification in figure 5.2

Figures 5.1 and 5.3 show that gear meshes with lower pressure angles and higher number of teeth exhibit lower transmission enor. The high density of lines that start at the lower left comer and proceed upwards represents a transitional region between high contact ratio (HCR) and low contact ratio gears (LCR). The optimum point P represents 76 a high contact ratio gear design. Note that in the low contact ratio region, the function is less sensitive to changes in pressure angle. Hence designers should design gears with higher pressure angle if the manufacturing process shows considerable variation in pressure angle.

Table 5.1. Gear Geometry

GEOMETRIC PARAMETERS VALUES Transmitted torque (in-lbs) 1250.0 Center Distance (inch) 2.7953 Gear Ratio 1.000 Number of teeth 58.21 Normal diametral pitch (1/in) 16.6323 Normal pressure angle (deg) 11.3643 Outer diameter (inch) 3.6202 Root diameter (inch) 3.3489 Profile contact ratio 2.5077 Roll angle @ pitch circle 11.516° Roll angle @ tip circle 19.269° Roll angle @ SAP 3.762° Roll angle® HPDTC 16.130° Amount of Tip Relief (inch) 0.0004 Amount of Lead modification (inch) 0.0004

5.3 Example #2 The first example illustrated our method using a gear mesh with unity gear ratio. This example will illustrate a design of a spur gear mesh for non-unity gear ratio. When the pinion and the gear are standard designs with gear having the larger number of teeth than the pinion, the pinion tends to be weaker than the gear due to its fillet geometry. In the gearing industry, engineers usually try to balance the strength of the pinion and the gear. 77 There are various ways one could design gear meshes with strength balanced between their members. One such way is by applying addendum modification (pulling or pushing the hob cutter) where the designer will pull the hob cutter out for the pinion by the same amount the hob cutter is pushed in for the gear. Such designs are refened to as Long addendum pinion meshing with short addendum gear(or also called Long-Short addendum mesh). This is equivalent to increasing the tooth thickness for the pinion by the same amount the tooth thickness is decreased for the gear. This example deals with the effect of hob shift on tooth strength and transmission enor.

The first part of this section will provide a brief discussion on the effect of hob shift in Long-Short addendum mesh on transmission enor and gear tooth strength. Second part will provide an optimization example where the goal is to minimize transmission enor for spur gear meshes with balanced strength.

5.2.1 Effect of Hob shift Figures 5.4 and 5.5 shows contour plots of equal PPTE for various combinations of hob shift coefficient and pressure angle for 40 tooth and 30 tooth pinion respectively. The hob shift coefficient was varied between 0 and 1.0 and pressure angle was varied between 15 and 25. From Figs. 5.4 and 5.5, one finds that for a given pressure angle,

PPTE is the lowest when there is no hob shift. The region identified by HCR represents designs with high profile contact ratio. Designs that use hob shift to balance the strength tend to have lower contact ratios than those that do not use hob shift. The dark curve that runs from left to right in Figs. 5.4 and 5.5 indicates the amount of profile shift required to balance the strength of the pinion and the gear. The sudden discontinuity in this curve is due to the use of a discrete square grid (1 lxl 1) of data points for contour plotting. 78 1.0

C/5 I

0.0 15 Pressure Angle in degrees 25 Fig. 5.4 Contours of equal PPTE for various gear meshes defined by hob shift coefficient and pressure angle for 40 teeth in the pinion

1.0

c .a

**H 8 3 OO X)

0.0 15 Pressure Angle in degrees 25 Fig. 5.5 Contours of equal PPTE for various gear meshes defined by hob shift coefficient and pressure angle for 30 teeth in the pinion 79

To confirm the results obtained in Figs. 5.4 and 5.5, a LCR spur gear design was selected with following specifications and design parameters: 1) Specifications: Transmitted torque of 1250; center distance of 3.5 inches; gear ratio equals 2.5 and face width of 1 inch.

2) Design parameters: 30 tooth pinion; normal generating diametral pitch equals 15; helix angle equals 0° and normal generating pressure angle equals 20°.

600

-D SHIFT=0.5 -o^SHIFT=0.4 -O SHIFT=0.3 D- — SHIFT=0.2 o- — SfflFT=0.1 * — SfflFT=0

ONOOt^voin-tfrrifS-HOv O\0\0\0\0\0\0\0\0\ S -HCSri-iTj-iovot^-oo^ ooooooooo Number of Mesh Cycles

Fig. 5.6 Transmission Enor v/s number of mesh cycles for a LCR spur gear with various hob shift coefficients

Fig. 5.6 plots the transmission enor versus gear rotation for all the six values of hob shift. Note that the difference in the plots is small and PPTE is insensitive to changes in hob profile shift. However, there is a small decrease in the profile contact ratio as one increases hob shift. One cannot achieve significant decrease in PPTE by designing 80 Long-Short addendum mesh and the primary purpose of Long-Short addendum mesh in spur gears is to balance the strength of the pinion and the gear.

Figures 5.7 and 5.8 deal with helical gear designs. Fig 5.7 plots the transmission enor over one mesh cycle for a helical gear mesh with face contact ratio equal to one. The helical gear design shown in Fig. 5.7 used the following specifications and geometric design parameters:

1) Specifications: Transmitted torque of 1250; center distance of 5.2725 inches; gear ratio equals 4.0 and face width of 1 inch. 2) Design parameters: 20 tooth pinion; normal generating diametral pitch equals 10; helix angle equals 18.5° and normal generating pressure angle equals 20°.

490

-6 SHIFT=0.5 -X SHIFT=0.4 -n SHIFT=0.3 o---SHIFT=0.2 D- — SHIFT=0.1 O SHIFT=0.

I I I I I I I I I I I I I 1 I I I I I I ON oo r> vo

Fig. 5.7 Transmission Enor v/s number of mesh cycles for a helical gear (FCR=1) with various hob shift coefficients 81 Fig. 5.8 plots the transmission enor over one mesh cycle for a helical gear mesh with face contact ratio greater than one. The helical gear design in Fig. 5.8 used the following specifications and geometric parameters: 1) Specifications: Transmitted torque of 1250; center distance of 5.5168 inches; gear ratio equals 4.0 and face width of unity. 2) Design parameters: 20 tooth pinion; normal generating diametral pitch equals 10; helix angle equals 25° and normal generating pressure angle equals 20°.

-SHTFT==0. 5

X— -SHIFT= =0.4

-SHTFT==0. 3 o — -SHIFT= =0.2

••_ -SHTFT==0. 1 —o-— -SHIFT= =0. H 410 ro «S -H Ov OsQsOsOsOsOsOsOr» vp

Fig. 5.8 Transmission Enor v/s number of mesh cycles for a helical gear (FCR > 1.0) with various hob shift coefficients

Though there exists differences in the transmission enor plot in Figs. 5.7 and 5.8 for the various hob profile shifts, PPTE value does not vary significantly for different values of hob shift.. Face contact ratio in helical gears does not change with hob profile shift while the profile contact ratio does vary in small values. As changes in PPTE for various hob shifts in Long-Short addendum mesh are negligible, one can conclude that primary 82 purpose of hob profile shift is to balance the bending strength of the pinion and the gear. However, there are also other reasons for designing Long-Short addendum gear meshes. In speed reduction gear drives, gear engineers apply a positive hob shift (pull) for the pinion and a negative hob shift (push) for the gear to increase the time during which mesh is in recess action. Such designs reduce specific sliding near the root of the pinion and hence increase its durability. In many cases, due to a space constraint, designers choose gears with smaller number of teeth and in such cases, positive hob shifts are used to avoid undercut at the root of the pinion tooth.

5.2.2 Optimization example This section illustrates the design of a spur gear pair with balanced strength and lower transmission enor. The specifications for this example are as follows: 1) Transmitted torque of 1000 in-lbs 2) Center Distance of 3.5 inches 3) Gear ratio of 2.5

During optimization, the following variables were varied:

1) Number of teeth on the pinion (Np) was continuously varied between 15 and 75.

2) Normal Generating pressure angle (§n) was continuously varied between 10 and 25 degrees. 3) Hob shift coefficient was varied between 0.0 and 1.0

The following parameters were assumed during optimization: 1) The face width equals 1 inch for both the pinion and the gear.

2) The tooth thicknesses of the pinion and the gear were reduced by 0.0025/Pnd at the standard pitch diameter to account for positive backlash. 83 3) The hob shift coefficient was equal in magnitude for both the pinion and the gear. However, positive hob shift was applied for the pinion while negative hob shift was applied for the gear. 4) This example used standard hobs with hob addendum coefficient of 1.25 and hob tip radius coefficient of 0.157.

Following were the constraints modeled during optimization: 1) Minimum number of teeth to avoid undercut. This is a function of pressure angle, hob shift, hob addendum and hob tip radius.

2) Minimum top land thickness for the pinion of 0.3/Pnd where P„d is the normal generating diametral pitch.

6) The bending stress of the gear should be equal to that of the pinion. This infact

makes the set of constraints #4, #5 and #6 dependent on one another. 7) The contact stress at the lowest point of single tooth contact on the pinion should be less than 150 ksi.

Table 5.2 tabulates the geometric parameters that represent the gear mesh given by the results of optimization. The results show that the pinion and the gear have balanced bending stress. In Figs. 5.1 and 5.3, the optimum was located in the high contact ratio region. In this example, the bending stress of the pinion and the gear limits the design from lying in the high contact ratio region. For the optimum to be located in the high 84 contact ratio region, the optimum design should have higher number of teeth which would decrease the size of the gear tooth and thus increase its bending stress.

When compared to the spur gear designs shown in section 4.5, gear meshes treated in this example are heavily loaded. This is one of the reasons why the optimization process converges to an optimum where the pressure angle is very high.

Table 5.2 . Gear Geometry for the Optimum

DESIGN PARAMETERS VALUES Transmitted torque (in-lbs) 1000.0 Center Distance (inch) 3.5 Gear Ratio 2.5 Number of teeth on the pinion/gear 28.9/72.25 Normal diametral pitch (1/in) 14.4494 Normal pressure angle (deg) 25.0 Outer diameter of the pinion/gear (inch) 2.1572/5.1196 Root diameter of the pinion/gear (inch) 1.8454/4.8078 Pinion trans, tooth thick @ op. pit. dia. (inch) 0.11731 Gear trans, tooth thick @ op. pit. dia (inch) 0.09977 Profile contact ratio 1.5063 Bending stress of the pinion/gear (psi) 30005/29991 Contact Stress at pinion LPSTC (psi) 142247 PPTE for unmodified tooth (u inches) 191.91

In practice, it is infeasible to have non-integer number of teeth. Based on the results, a designer will choose 29 teeth for the pinion and 72 teeth for the gear. Such an approximation would shift the gear ratio from 2.5 to 2.48. However, the designer usually has some tolerance on the gear ratio to work with. The purpose of this example was to 85 study the effect of design variables on transmission enor and hence no approximation was done to convert the number of teeth to an integer in table 5.2.

5.3 Example #3 This section illustrates the design of a helical gear pair with lower transmission enor. by solving two design problems. Design problem 1 involves minimizing transmission enor for standard gear meshes without any hob shift. Design problem 2 involves Long- Short addendum gear designs to not only minimize transmission enor but also to balance the bending strength between the pinion and the gear. The specifications for problems 1 and 2 are as follows:

1) Transmitted torque of 1000 in-lbs 2) Center Distance of 2.7953 inches

3) Gear ratio of 2.75

During optimization, the following variables were varied:

1) Number of teeth on the pinion (Np) was continuously varied between 15 and 75.

2) Normal Generating pressure angle (tyn) was continuously varied between 10 and 25 degrees. 3) Helix angle was varied between 0.0 and 45 degrees.

4) In design problem 1, the hob shift coefficient has a constant value of zero while in design problem 2, it is varied continuously between 0.0 and 1.0.

The following parameters were assumed during optimization: 1) The face width was assumed as unity for both the pinion and the gear.

2) The tooth thicknesses of the pinion and the gear were reduced by 0.0025/Pnd at the standard pitch diameter to account for positive backlash. 86 3) Design problem 1 considered only standard gears and hence the operating pitch diameter was the same as the standard pitch diameter. The addendum coefficient of the pinion and the gear was 1.0. Design problem 2 applied hob shift in equal magnitudes for both the pinion and the gear. However, positive hob shift was applied for the pinion and negative hob shift was applied for the gear. 4) Design problems 1 and 2 used standard hobs with hob addendum coefficient of 1.25 and hob tip radius coefficient of 0.157.

Following were the constraints modeled during optimization: 1) Minimum number of teeth to avoid undercut. This is a function of pressure angle, helix angle, hob shift, hob addendum and hob tip radius.

2) Minimum top land thickness for the pinion of 0.3/Pnd where Pnd is the normal generating diametral pitch.

3) Minimum top land thickness for the gear of 0.3/Pnd- 4) Bending stress of the pinion should be less than 30 Ksi. The procedure evaluated the bending stress using the AGMA geometry factor method (Erichello ,1981). 5) Bending stress for the gear should be less than 30 Ksi. 6) The contact stress at the operating pitch radius should be less than 150 ksi. 7) For design problem 2, the bending stress of the gear was required to be equal to that of the pinion.

Table 5.3 tabulates the design parameters that represent the gear mesh given by the results of optimization for both problems 1 and 2. The importance of face contact ratio in helical gears can be seen in the results. The optimum design exhibits a face contact ratio of 1.99556. When helical gear meshes have integral face contact ratio, the total 87 length of lines of contact remains constant over a mesh cycle. This phenomenon reduces amount of mesh stiffness variation and hence lowers variation in transmission enor.

Table 5.3 . Gear Geometry for the Optimum

GEOMETRIC PARAMETERS UNBALANCED BALANCED Transmitted torque (in-lbs) 1000.0 1000.0 Center Distance (inch) 2.7953 2.7953 Gear Ratio 2.75 2.75 Number of teeth on the pinion/gear 25.13/69.11 26.29/72.3 Normal diametral pitch (1/in) 17.984 18.715 Normal pressure angle (deg) 29.056 28.555 Helix Angle (deg) 20.4019 19.5668 Outer diameter of the pinion/gear (inch) 1.6020/4.2110 1.6184/4.1860 Root diameter of the pinion/gear (inch) 1.3516/3.9605 1.3777/3.9453 Pinion trans tooth thick @ op. pit. dia. (inch) 0.09304 0.10087 Gear trans tooth thick @ op. pit. dia. (inch) 0.09304 0.07699 Profile contact ratio 1.2799 1.2889 Face contact ratio 1.9956 1.9951 Total contact ratio 3.27535 3.2839 Bending stress of the pinion/gear (psi) 29998/27580 30356/30391 Contact stress at op. pit. dia (psi) 147894 148587 PPTE for unmodified tooth (\i inches) 10.80 11.07

Helical gears are not as prone to undercutting as spur gears. The two constraints that are active for the optimum are constraints #4 (pinion bending stress) and #6 (contact stress). The optimum value of PPTE for helical gears are much lower than that of spur gears because in helical gear meshes there are 2 or more pairs of teeth in contact. The change in mesh stiffness from one pair to 2 pairs of teeth in mesh will be gieater than the 88 change in mesh stiffness from 2 pairs to 3 pairs in contact. This is the same reason why high contact ratio spur gears show lower PPTE than low contact ratio spur gears.

5.5 Example #4 This example applies the techniques discussed in chapter 4 to the design of helical gears. The objective in this example is to design a helical gear mesh that has minimum transmission enor and at the same time, is not sensitive to manufacturing enors in pressure angle and helix angle. The specifications for this problem are: 1) Transmitted torque of 1000 in-lbs 2) Center Distance of 2.7953 inches 3) Gear ratio of 2.75

During optimization, the following variables are varied:

1) Number of teeth on the pinion (Np) is continuously varied between 15 and 75.

2) Normal Generating pressure angle (n) is continuously varied between 10 and 25 degrees. 3) Helix angle is varied between 0.0 and 45 degrees.

The following parameters are assumed during optimization: 1) The face width is assumed as unity for both the pinion and the gear.

2) The tooth thicknesses of the pinion and the gear are reduced by 0.0025/Pnd at the standard pitch diameter to account for a positive backlash.. 3) This example considers standard gears with the operating pitch diameter equal to the standard pitch diameter. The addendum coefficients of the pinion and the gear are 1.0. 89 4) This example uses standard hobs with hob addendum coefficient of 1.25 and hob tip radius coefficient of 0.157. 5) The manufacturing enors in helix angle and pressure angle are 0.2 and 0.4 degrees respectively.

The following are the constraints modeled during optimization: 1) Minimum number of teeth to avoid undercut. This is a function of pressure angle, helix angle, hob shift to account for backlash, hob addendum and hob tip radius.

2) Minimum top land thickness for the pinion of 0.3/Pnd where Pnd is the normal generating diametral pitch.

3) Minimum top land thickness for the gear of 0.3/Pnd- 4) Bending stress of the pinion should be less than 30 Ksi. The procedure evaluated the bending stress using the AGMA geometry factor method (Enichello ,1981). 5) Bending stress for the gear should be less than 30 Ksi. 6) The contact stress at the operating pitch radius should be less than 150 ksi.

Table 5.4 shows the results of optimization. A unique combination of three design variables represents a target design. The procedure evaluates PPTE at the target design and at the 4 worst combinations of design variables (due to enors in pressure angle and helix angle) given by L4 OA. Equations 3.8 and 3.7 evaluated the sensitivity index and the objective function respectively. The weighting factor (a) was assumed as 0.5. The optimum design shown in Table 5.4 exhibits a near integer face contact ratio mesh. When compared with designs shown in Table 5.3, this target design has lower bending stress because the target design point has moved away from the stress constraint to avoid any constraint violation (higher stress) by the designs represented by the worst combinations of design variables. Table 5.5 shows the PPTE values for the target and at 90 the 4 worst combinations of helix angle and pressure angle. Table 5.6 gives the variance caused by each design variable. The results in Table 5.6 show that PPTE is very sensitive to enors in helix angle and insensitive to enor in pressure angle. Hence during the manufacturing process (molding process, etc.), engineers should try to control process parameters that affect helix angle.

Table 5.4. Gear Geometry for the target design

GEOMETRIC PARAMETERS VALUES Transmitted torque (in-lbs) 1000.0 Center Distance (inch) 2.7953 Gear Ratio 2.75 Number of teeth on the pinion/gear 23.89/65.7 Normal diametral pitch (1/in) 17.189 Normal pressure angle (deg) 29.4293 Helix Angle (deg) 21.2090 Outer diameter of the pinion/gear (inch) 1.6072/4.2161 Root diameter of the pinion/gear (inch) 1.3451/3.9541 Transverse backlash (inch) 0.0003 Percentage backlash on pinion 50. Profile contact ratio 1.2606 Face contact ratio 1.9794 Total contact ratio 3.2400 Bending stress of the pinion/gear (psi) 28421/26047 Contact Stress at op. pit. dia (psi) 148395 PPTE for unmodified tooth (|j, inches) 13.2152

This example applied constrained statistical optimization techniques to a helical gear design with variations in pressure angle and helix angle and showed that transmission 91 enor is very sensitive to enors in helix angle and less sensitive to pressure angle at higher pressure angles. Table 5.5 PPTE at all worst combinations

Trial 0n Vn PPTE Target 29.4292 21.2090 13.2152 lof4 29.8293 21.4090 11.5193 2 of 4 29.0293 21.0090 14.3482 3 of 4 29.8293 21.0090 14.5913 4 of 4 29.0293 21.4090 11.1173

Table 5.6 Sum of squares

Source Sum of squares All factors (Total) 10.04 n 0.104 9.931

5.6 Example #5 In the gear meshes used in sections 5.2, 5.3, 5.4 and 5.5, gears had standard addendum coefficient (=1.0) and operated at standard center distances. The normal generating diametral pitch was a function of number of teeth and specified gear ratio and center distance. This example will discuss briefly the effect of designing gear meshes where the operating pitch diameter is not equal to the standard pitch diameter.

This example varies the following two variables for a given value of number of teeth, pressure angle, helix angle, gear ratio and center distance: 92 1) Normal generating diametral pitch by small amount from its nominal value (determined by center distance, number of pinion teeth, helix angle and gear ratio) 2) Working depth (sum of the addendums of the pinion and the gear) by changing the outside diameter (or addendum coefficient).

Fig. 5.9 shows contour plots of equal PPTE for various values of normal generating diametral pitch and working depth. The specifications and design parameters used for gear meshes in Fig. 5.9 are as follows: 1) Specifications: Transmitted torque of 1000 in-lbs; center distance of 4.2; gear ratio of 2.5 and facewidth equals 1.0 for both the pinion and the gear. 2) Design parameters: 24 teeth on the pinion; normal pressure angle of 20 degrees and helix angle of 0 degrees

From the specifications and the assumptions used, the standard value for the diametral pitch is 10.0. In Fig. 5.9, the diametral pitch is varied between 10 and 10.6 and the working depth was increased from 2.0 to 2.4. The profile contact ratio (PCR) increases as the outside diameter is increased for a given value of normal generating diametral pitch. It is important to note that PCR is a maximum when the diametral pitch is the smallest and the working depth is the highest. As diametral pitch increases, the standard pitch diameter decreases. This example pulls the hob for both the pinion and the gear by an amount equal to the difference between the standard pitch diameter and the operating pitch diameter. As the center distance and gear ratio remain constant, the operating pitch point remains the same. The strength of the pinion and the gear are higher due to a positive hob shift. In Fig. 5.9, gear meshes with higher diametral pitch have lower stresses. Fig 5.9 graphs contours of equal PPTE for various gear meshes with unmodified teeth and shows that one can minimize transmission enor in spurs gears by varying the 93 working depth and diametral pitch. The optimum design has a higher working depth and a higher diametral pitch than the standard design (diametral pitch =10.0, working depth = 2.0). However, occunence of pointed teeth and root interference limit the amount by which one can vary working depth and diametral pitch to minimize transmission enor.

2.0 Working Depth 2.4

Fig. 5.9 Contours of PPTE in microinches for various spur gear meshes

In Fig. 5.9, all gear meshes have profile contact ratio less than 2.0 (low contact ratio design). However, by varying working depth, if PCR had increased to a value greater than 2.0 (high contact ratio design), then these designs will exhibit lower transmission enor values than the low contact ratio designs. Hence in spur gears, designers can minimize transmission enor by varying diametral pitch and working depth. 94 Fig 5.10 shows the effect of normal generating diametral pitch and working depth on transmission enor in helical gears. The specifications used for gear meshes in Fig. 5.10 are as follows: 1) Specifications: Pinion torque of 1000 in-lbs; center distance of 4.4695"; gear ratio of 2.5 and facewidth equals 1.0 for both the pinion and the gear. 2) Design parameters: 24 teeth on the pinion; Normal pressure angle of 20 degrees and Helix angle of 20 degrees

2.0 Working Depth 2.4 Fig. 5.10 Contours of PPTE for various helical gear meshes

Figure 5.10 shows that PPTE is minimum for gear designs with coarser diametral pitches and higher working depth. The total contact ratio for all gear meshes is between 2.0 and 3.0. The face contact ratio (FCR) does not vary for different working depth but does vary with different diametral pitches. The profile contact ratio is larger for lower 95 diametral pitches and higher working depth. The face contact ratio is higher for larger diametral pitch. The amount by which PCR changes over the domain is more than the amount by which FCR changes. Hence the maximum total contact ratio occurs for higher working depth and lower diametral pitch. In this example, PPTE is minimum where total contact ratio is maximum.

2.0 Working depth 2.4 Fig. 5.11 contours of equal PPTE for various helical gear meshes

In Fig. 5.10, varying diametral pitch from its nominal value does not reduce transmission enor. However, if the face contact ratio is an integer for any of these designs, then PPTE would be a minimum for that design. Fig. 5.11 plots contours of equal PPTE for a gear design where the face contact ratio changes from above 1.0 to below 1.0 when diametral pitch and working depth are varied. The specifications and design parameters used for gear meshes in Fig. 5.11 are as follows: 96 1) Specifications: Transmitted torque of 1000 in-lbs; center distance of 4.4161"; gear ratio of 2.5 and facewidth equals 1.0 for both the pinion and the gear. 2) Design parameters: 24 teeth on the pinion; Normal pressure angle of 20 degrees and helix angle of 18 degrees

Fig. 5.11 shows that one can reduce PPTE by varying diametral pitch and working depth in non-standard helical gear designs. Hence one can not infer that diametral pitch does not have improve transmission enor in helical gears. Due to higher contact ratio, increasing working depth does reduce transmission enor.

Figures 5.9 and 5.11 showed that one can reduce transmission enor in spur and helical gears by varying diametral pitch and working depth. However, there are other parameters like face and profile contact ratios that change with these parameters that affect PPTE. There is a need for a detailed study on how working depth and diametral pitch in non-standard gear meshes affect transmission enor.

5.7 Summary This chapter illustrated the effect of number of teeth, diametral pitch, pressure angle, helix angle, hob shift and working depth on transmission enor. Some of the conclusions made in this chapter verifies the traditional knowledege in gear engineers gained over years by experience. The following conclusions can be drawn from the examples shown for each design variable:

1) Number of Teeth: The higher is the number of teeth, lower is the transmission enor. However, the bending stress constraint limits the maximum number of teeth and undercut constraint gives the minimum number of teeth. 97 2) Diametral pitch: In standard gear designs, diametral pitch is directly related to number of teeth. Hence in such designs, higher the diametral pitch, lower is the transmission enor. In non-standard gear designs, diametral pitch is varied in small amounts. Example 5 showed that in non-standard spur gears one can minimize transmission enor in spur gears by varying diametral pitch.

3) Pressure angle: The lower is the pressure angle, higher is the contact ratio and hence lower is the transmission enor. In high contact ratio designs with standard working depth, designer should choose lower pressure angle. However lower pressure angle gears are susceptible to undercutting at the root of the gear tooth. When high contact ratio spur designs with standard working depth are infeasible due to large loads, designers can increase working depth to increase contact ratio. If high contact ratios are infeasible, one should prefer larger pressure angles as PPTE is insensitive to enors in pressure angle at higher pressure angles.

4) Helix angle: One should choose helix angle such that face contact ratio is an integer. Gear meshes with integer face contact ratios have constant length of lines of contact and hence lower PPTE. Gears with face contact ratio of 2.0 have lower transmission enor than gears with face contact ratio of 1.0.

5) Hob shift: In Long-Short addendum gear meshes, use of hob shift balances the strength of the pinion and the gear but does not show a significant effect on transmission enor.

6) Working depth: Increasing working depth does increase profile contact ratio and hence decreases PPTE. CHAPTER VI

CONCLUSIONS

6.1 Concluding Remarks This dissertation proposed a methodology to incorporate manufacturing tolerances and operational variances in the design optimization stage to achieve designs with robust and optimal performance. The dissertation applied the proposed methodology to design of spur and helical gears that have minimum transmission enor and at the same time, are less sensitive to shaft misalignment, torque variations and manufacturing enors in profile, pressure angle and helix angle.

Chapter 1 presented a brief introduction to concepts in Design for Robustness (DFR) and to the design of gears with minimum transmission enor. The emphasis was on a need for a methodology that incorporates manufacturing and operational variances in the design optimization stage to achieve robust and optimal performance. The chapter also discussed various methods reported in the literature.

Chapter 2 dealt with minimizing transmission enor in spur and helical gears using tooth modifications. The focus was on the merits and demerits of following three types

98 99 of tooth modifications: 1) Conventional profile and lead modification, 2) Cross modification and 3) 3-D topographical modification. Conventional profile and lead modification minimize transmission enor but show higher load distribution factors. Cross modifications are very sensitive to enors in profile modification and variations in torque. One can design zero transmission enor gear meshes using 3-D topological modifications but they are difficult to manufacture with the cunent technology.

Chapter 3 dealt with unconstrained statistical optimization. The proposed procedure uses concepts from statistical design of experiments to evaluate the expected value of a performance function over the tolerance space. The use of fractional factorials reduced number of performance evaluations during optimization. A helical gear example served to illustrate the use of the procedure.

Chapter 4 dealt with constrained statistical optimization which incorporates variations in constraints due to the variations in design variables. This chapter proposed three solution techniques: 1) Brute force with heuristics, 2) Constraints with built-in constraint variation, and 3) Approach using KKT conditions. The three techniques were illustrated using a spur gear example. The first two techniques are ideal for problems where function evaluation is expensive and constraint evaluation is inexpensive. The approach using KKT conditions is ideal for problems in which constraints and function evaluations are computationally expensive.

Chapter 5 studied the effect of geometrical design variables on transmission enor using illustrative examples. The variables were variables number of teeth, pressure angle, helix angle, diametral pitch, hob shift and working depth. The analysis addressed constraints such as minimum number of teeth to avoid undercut, maximum bending and 100 contact stress. The resulting conclusions include: 1) Higher the profile contact ratio, lower the transmission enor, 2) Higher the number of teeth, lower is the transmission enor, and 3) Face contact ratio should be close to an integer. Effect of designing Long- Short addendum gear meshes was also studied.

6.2 Contributions of this Research The major contributions of this dissertation are in the following two fields: 1) Statistical optimization 2) Gear design with minimum transmission enor

6.2.1 Contributions in statistical optimization Previous work in this area addressed only problems with analytical objective functions in closed-form equations. Their approach was not readily applicable to problems involving objective functions that are expensive to evaluate. This dissertation proposed practical solutions to these problems by developing the following concepts:

1) Use of design of experiments: The statistical optimization technique developed in this research applies statistical design of experiments concepts in the computation of the expected value of a performance function. The developed

technique is very effective when time-consuming computer models are used to measure performance for design problems with many variables. In section 3.5, the method reduced number of function evaluations by half compared to our previous technique by using an L8 orthogonal anay to model variations in 4 design variables. 101 2) Index of sensitivity: The procedure defines a measure of sensitivity called Sensitivity Index(SI) which is a root mean square value of the function values at the worst combinations of design variables. This concept is readily applicable to any problem that involves the measure of sensitivity. This concept models the effect of interactions between design variables where as concepts that use gradients to measure sensitivity do not.

3) Constrained optimization: This research proposed three techniques to solve constrained statistical optimization problems and defined various conditions of criticality of a constraint. The three statistical optimization techniques not only incorporate the effect of variations in design variables(AXj), but also the uncertainty in the constraints(Agj) due to variations in design variable.

The developed statistical optimization methodology can be readily applied to other engineering design problems such as profile design for beverage can bottoms, etc. Potential applications also include design of rotational plastic parts and heat treated shaft elements.

6.2.2 Contributions in gear design Previous work in this field did not address the need for designing spur and helical gears that not only have minimum transmission enor but also are insensitive to manufacturing enors, shaft misalignment and torque variations. This dissertation addressed this need. The major contributions are:

1) Effect of tooth modifications: This research identified the merits and demerits of the following three types of tooth modifications: 1) Conventional profile and lead 102 modification, 2) Cross modification and 3) 3-D topographical modification. Chapter 2 dealt with minimizing transmission enor and its sensitivity to torque variations, misalignment and enors in tooth profile.

2) Effect of geometric design variables: The proposed method enables designers to achieve spur and helical gears with lower vibration and longer life. Geometric constraints like avoiding undercut and maximum bending and contact stress were used during optimization procedure. This research identified the effects of pressure angle, helix angle, number of teeth, hob shift and working depth and parameters like contact ratios on transmission enor. Enors in pressure angle and helix angle were modeled during statistical optimization procedure. Traditionally, engineers selected values for geometric design variables based on bending and pitting (contact stresses) strength. This dissertation selected values for design variables based on transmission enor and used bending and pitting strength issues as constraints during the design process.

A gear design program based on the developed methodology can help designers in generating gear mesh designs quicker and with lower transmission enor.

6.3 Recommendations for Future Work The statistical optimization procedure assumed that variations in design variables are statistically independent. However, there are manufacturing processes like heat treatment where the variation in the design parameters are not independent. Future studies in Design for Robustness should address the need for incorporating such issues in the optimization procedure. Other techniques used to evaluate sensitivity derivatives 103 (Vanderplaats , 1985 and Beltracchi, 1988a) can also be incorporated into the statistical optimization procedure.

In the area of gear design, this dissertation studied the effect of geometric design variables on transmission enor. However, most of the examples used were gear designs operating at standard center distances. The effects of diametral pitch and working depth on contact ratios and their effect on transmission enor need to be studied in detail.

Not all gear design problems have a fixed center distance. In many problems, designers try to minimize center distance in order to minimize space and weight. One can extend the methods discussed in this work to such problems. There are also other constraints like scoring criteria that need to be incorporated in the optimization process.

Cunently, LDP assumes the gear tooth contact to occur along the line of action. When gear teeth do not have sufficient tooth modification, tooth contact does deviate (comer contact) from the line of action. One needs to upgrade LDP to model comer contacts.

This dissertation characterized transmission enor by its peak to peak value. One can also characterize transmission enor by the Fourier amplitude at the meshing frequency and its multiples. Subsequently, there is a need for studying the effect of geometric design variables on the amplitude of the transmission enor at the first few multiples of the meshing frequency.

This dissertation assumed uniform load distribution while evaluating bending stresses using the geometry factor method. One can also evaluate stresses using the load 104 distribution predicted by LDP. Work is cunendy being conducted at the Ohio State University to develop a methodology to evaluate bending stresses using the load distribution predicted by LDP.

The manufacturing enors in the tooth profile depends on the machine tool and the manufacturing process. One can specify the manufacturing tolerance in the gear tooth profile in different ways and subsequently, there is a need to study the effect of different methods of specifications on the sensitivity of transmission enor.

Finally, one should study the sensitivity of transmission enor to enors in manufacturing and operation experimentally. BIBLIOGRAPHY

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INTRODUCTION TO ANALYSIS OF VARIANCE

A.1 Introduction

Analysis of variance (ANOVA) is a technique that breaks the total variation into accountable variations. The total variance (SSj) is divided into the following variations: l)Variance due to main effects (individual effects of factors like A, B, etc.), 2) Variance due to interactions (combined effect of two factors or more). This section introduces analysis of variance with an example. Calculations similar to the ones used in section 3.5 are shown. Detailed descriptions of this method can be found in Scheffe' (1959), Ross (1988) and Box (1961 a and b).

A.2 Illustration The following example uses a L8 orthogonal anay to explain this technique. An L8 orthogonal anay is used to analyze the effects of four factors (A, B, C and D) and interaction AB. Each row in the L8 anay is replicated so that we have two trials for each selected combination of factors. Factors A, B, C and D are assigned to columns 1, 2, 4 and 7 respectively. For example, in the two trials of the first experiment, all factors will

110 Ill be in their first level where as in the two trials of the 4th experiment, factors A and D will be in their first level and factors B and C will be in their second level. Table A.l presents the data for this example.

Table A.l Illustrative Data

Columns Response Yi Expt. No A B C D Trial #1 Trial #2 1 1 1 1 1 1 1 1 9 10 2 1 1 1 2 2 2 2 11 10 3 1 2 2 1 1 2 2 13 11 4 1 2 2 2 2 1 1 12 11 5 2 1 2 1 2 1 2 15 14 6 2 1 2 2 1 2 1 13 12 7 2 2 1 1 2 2 1 16 17 8 2 2 1 2 1 1 2 18 16

Let T be total of all responses Yj, and N be the total number of trials . Let Aj and A2 be the sums of the data associated with the first and second levels of factor A. Similarly, Bj and B2 are the sums of data associated with the first and second levels of factor B and so on for other factors. For the given data, N=16 N T=2Yi=208 i=l Ai = 9+10+11+10+13+11+12+11 =87

A2 = 15+14+13+12+16+17+18+16 = 121 B, = 9+10+11+10+15+14+13+12 = 94

B2 = 13+11+12+11+16+17+18+16 =114 112

C2 =9+10+13+11+15+14+16+17 = 105

C2 =11+10+12+11+13+12+18+16 = 103

Dt = 9+10+12+11+13+12+16+17 = 100

D2= 11+10+13+11+15+14+18+16= 108 (A.l)

Let n^j and n^2 represent the number of observations when A was at level 1 and level 2 respectively. Similarly nBi, nB2, n^i, n^2, nDi and nD2 represent number of observations for factor B at level 1, factor B at level 2, factor C at level 1, factor C at level 2, factor D at level 1 and factor D at level 2 respectively, n^; represent number of observations for when factor A is at level i and factor B is at level j. nAl=nA2=nBl=nB2=nCl=nC2=nDl=nD2=8 nAlBl=nAlB2=nA2Bl=nA2B2=4 (A-2)

To estimate the effect of each factor, we calculate parameters called sum of squares. These are nothing but the sum of the squared deviations from the mean. The simplified formulae for sum of squares are as follows:

Grand total sum of squares (SSj) is the sum of squared deviations from the overall mean m (=T/N) and is given by

N T2

Sum of squares due to main effect A (SSA) is given by

A2 A2 T2 N "A, "A2 (A.4) Similarly SSn, SSQ and SSD are given by

Tj2 TJ2 T<2 ssB=^-+l?—-L (A.5)

r^Z f*\L *T»^ SS = -^-+-^—— c n- n N c. "cn> " (A.6)

2 2 n D2 T SSD = -^+-^-— n n N D, D3 (A.7)

Sum of squares due to interaction(AB) is given by

2 2 2 2 2 SSAB -(A^) , (AtB2) ^A.B,) ^A.B,) T ggA _SSB

n n n n N A,B, A,B2 A2B, A2B2 (A.8)

where (AjBp is the sum of the observations(responses) when factor A is at level i and factor B is at level j. For the given data (AiBi) = 9+10+11+10 = 40

(AiB2)= 13+11+12+11 =47 (AiBi)= 15+14+13+12 = 54 (A^) = 16+17+18+16 = 67 (A.9)

SST=112

SSA = 72.25 (65%ofSST)

SSB = 25.0 (22%ofSST)

SSC = 0.25 (0.22%ofSST) 114

SSD = 4.0 (3.6%ofSST)

SSAB = 2.25 (2.0 % of SST) (A. 10)

For the above results, one can say that factor A has the most significant effect and factor C has the least significant effect.

The concept of matrix experiment discussed so far is used to evaluate the effects of each factor on performance f. Based on the analysis, one can determine those factors that influence the performance most and choose levels(or values) for the design variables(or process parameters) such that performance is optimum. For example, if performance f is the largest at level A2, and the objective is to minimize performance f, then one should choose Al for factor A.. When factors are modeled with two levels, linear effects of factors can be studied, ie. performance is either monotonically increasing or decreasing with the levels of that factor. By choosing three levels for each factor, we can study the non-linear effects of each individual factors. Taguchi has proposed 3-level anays that can be used to study the non-linear effects of design variables. APPENDIX B

ALGORITHMS USED IN OPTIMIZATION SCHEME

B.l Introduction This appendix gives a brief description of the techniques used during optimization. Section B.2 explains the Golden section method used for the one dimensional search algorithm. Section B.3 presents Broydon-Fletcher-Goldfarb-Shanno (BFGS) variable metric method used for unconstrained minimization. Section B.3 discusses the Augmented Lagrange Multiplier method used for solving constrained optimization problems.

B.2 Golden Section Method The golden section method is one of the widely used methods for one-dimensional search during optimization. The function F is assumed to be unimodal. Vanderplaats (1984) gives a detailed description of this method. This section only outlines the steps involved.

Given x\: Lower bound for x

115 116

xu: Upper bound for x When the bounds are unknown, they can be found by using the algorithm presented later in this section. Fi: Function at xi.

Fu: Function at xu N: Maximum number of iterations x: Parameter used , value of x equals 0.381966 Stepl

xi = (l-x)xi + xxu (B.l) Evaluate function f at xt. Fi = F(xi)

x2 = x xi + (1-x ) xu (B.2)

Evaluate function f at x2. F2 = F(x2) K=3 Counter for number of iterations Step 2 K = K+1 If(K>N) Then Exit Step 3

If (Fi > F2) Then xi = xi

Fi = Fi

xi = x2

Fi = F2

x2 = x xi + (1-X ) xu

F2 = F(X2) Go to Step 2 Else 117

xu = X2

FU = F2

x2 = xi

F2 = Fi

xi = (1-x) xi + x xu Fi = F(xi) Go to Step 2 Endif (B.3)

The algorithm used to find bounds for x is as follows Stepl

Choose xl = 0.0 and xu arbitarily. Evaluate Fi = F(xj) and Fu = F(xu).

If (Fu > Fi) Then xu is the upper bound, Exit. Step 2

xi=xu

Fi=Fu

xu = (1+x) xi - x xi

If (xu > xmax) Then unbounded. Exit

Fu = F(xu)

If (Fu > Fi) Then xu is the upper bound, Exit. xl = xl F1 = F1 Go back to Step 2 (B.4) 118 B.3 BFGS Variable Metric Method BFGS variable metric method is one of the many first-order optimization procedures that utilizes the gradient information in its search strategy. These schemes usually calculate the gradient information using forward finite difference computations. The main advantage of BFGS method over other first-order methods is that it stores information about previous iterations in a matrix. Hence convergence occurs in lesser number of iterations. We describe the method in following steps:

Stepl Choose initial values for design variables x° and calculate the gradient, VF(x°) using forward finite difference.

Step 2. Determine search direction si for q* iteration using equation.(B.5)

sq = -KVF(xq) (B.5)

where K approaches the inverse of the Hessian matrix (H). The Hessian matrix is the matrix of second partial derivatives of the objective function with respect to design variables. For the initial design point, assume the K matrix to be an identity matrix.

Step 3 Using the search direction s% determine the new design point using equation (B.6).

xq+1=xq+asq (B.6) 119 In equation (B.6), choose a value of a such that F(xl+1) is a minimum value. If the optimum value of a is close to zero, then convergence criterion is satisfied. Otherwise, determine x

Step 4 Evaluate the change vectors (p and y) and scalars (o" and x).

p = xq+1 - xq (B.7)

y = VF(xq+1)-VF(xq) (B.8)

o = p.y (B.9)

x = yTKy (B.10) With calculated change vectors and scalars, evaluate the new K matrix using

q+ 1 q q Kq =Kq+Dq (B.ll)

In equation (B.l 1), Dl is given by

q T q T q T D = £+1 pp - I[K yp + p(K y) ] (B.12)

Once KQ+1 is known, go back to step 2 and perform the next iteration. 120 B.4 Augmented Lagrange Multiplier Method This section describes the steps involved in the Augmented Lagrange Multiplier (ALM) method used to solve constrained optimization problems. This method is also called as method of multipliers. Piene(1975), Vanderplaats (1984) and Reklaitis (1983) give a detailed description of this method. Only an outline is presented in this section.

Consider a constrained optimization problem

Minimize F(x) Subject to

gj(x)<0.0 j = l,m

hk(x) =0.0 k=l,p

where x= [xi, x2,... xn] (B.13)

The Lagrangian is given by equation (B.14)

m p L(x,*,p) = F(x) + 5>jgjW + £pkhk(x) (B.14) j=l k=l

where Xj and Pk are the Lagrangian multipliers associated with inequality and equality constraints. In the Augmented Lagrangian multiplier algorithm, a exterior penalty function is added to the lagrangian to create a psuedo-objective function (also called Augmented Lagrangian) given by equation (B.15). 121

111 t*p 2 A(xA,p,rp) = F(i) £[XjVj+rpv?] + £{pI k hk(x) + rp[hk(x)] } j=i k=r (B.15) where \|/j = max .,«.£

Piene (1985) provides a proof of a theorem that states that if A(x , X*, P*) is an unconstrained local minimum of A(x, X*, P*, rp) with respect to x for some finite rp > 0.0, then f(x*) is a constrained local minimum of f(x). The first step in this algorithm is to select values for X, P, and rp. The procedure then minimizes the augmented Lagrangian. Once the unconstrained optimum of A is obtained, the multipliers are updated for the q+lth unconstrained minimization using values used in the qth unconstrained minimization. The formulas for updating multipliers are given by equations (B.16) to (B.18)

+1 A] = X) + 2rpJ gj(x), j = l,m (B.16) max 2r„

+1 q + 2r_h (x) k = l,m K = P k k (B.17)

| xpmax ifyrp>r pmax lpnew r (B.18) lY n ifyr_

With the new values for X, p, and rp, augmented Lagrangian is now minimized till convergence criteria are met. The convergence criteria are as follows: if Fq+i-Fq| < ef then converged * * 1 if xq+l — xq| ^ ex then converged (B.19)

if kq+1 ~ ^q < EX then converged

In the equation (B.19), Ef, £x, and e^ represent the maximum allowable change in the value of objective function, design vector and Lagrange multipliers respectively.