Optimal Tolerance Allocation through Tolerance ...

International Journal of Applied Engineering Research, ISSN 0973-4562 Vol. 10 No.78 (2015) © Research India Publications; httpwww.ripublication.comijaer.htm

Optimal Tolerance Allocation through Tolerance Chain Identification System M. Thilak

N. Senthil Kumar

Faculty, Department of Mechanical Engineering TRP Engineering (SRM Group), Trichy, Tamilnadu, India [email protected]

Faculty, Department of Mechanical Engineering TRP Engineering (SRM Group), Trichy, Tamilnadu, India [email protected]

G. Jayaprakash Faculty, Department of Mechanical Engineering Saranathan Engineering College, Trichy, Tamilnadu, India [email protected] control tolerance buildup. In order to determine working dimensions and tolerances of a process plan, a tolerance chart is often employed. Tolerance chain identification is an important step involved in the tolerance charting procedures. The main purpose of the tolerance chain identification is to calculate the working dimension and tolerance of each operation.

Abstract— Tolerance plays an important role in the development and cost of manufactured products. The allocation of tolerance in the manufacturing process acts as a design tool to reduce total production cost, while satisfying the target quality level target. A graphical tool which is used to determine the working dimensions and their tolerances for manufacturing a product is called as tolerance charting. Tolerance chain identification is an important step in the tolerance charting. It is used to calculate the working dimension and tolerance of each operation. This paper presents a tolerance chain identification system based on arithmetic approach. A nonlinear optimization model is formulated with blue print tolerances, blue print dimensions and process capability as constraints which are used to calculate minimum manufacturing cost. Then the optimization model has been solved using Genetic Algorithm (GA).

II. LITERATURE REVIEW A wide range of research work is going on improving the tolerance charting process. Tolerancing based on a mathematical theory is proposed by Requicha. A. G [1]. It gives the basis for tolerancing to incorporate it into the Geometric solid modeling systems. Irani et al. [2] proposed a method for tolerance optimization. The various machine tools involved in the machining process, their machining capabilities and costs are considered for the formulation of constraints. Also, they assumed that the machining datums involved in the process are fixed. A graph representation method is proposed by Mittal et al. [3] to identify the dimensional chains. The graph obtained by this method includes the working dimensions only. Roy, U et al. [4] presented a technique to represent, manipulate and analyze the dimensioning and tolerancing data in CAD/CAM. The integration of CAD and CAM are also discussed. Clément et al. [5] proposed the concept of TTRS (Technologically and Topologically Related Surfaces).They created a hypothesis to derive inequalities concerning the maximum allowable values of parameters. A branch and link method is proposed by Ngoi and Fang [6] to present the process sequences in a table. They have created a tree diagram using the branches, leaves and linkers technique. Gauss elimination technique is applied to solve the linear equations which are created by

Keywords— Tolerance allocation; Tolerance charting; Tolerance chain; Genetic algorithm.

I. INTRODUCTION In today‟s competitive market place, cost savings and better performance plays a major role in the success of a product. So it is in need to produce products of good quality at low cost. One of the main key factor to consider about cost savings in design and manufacturing is mechanical tolerances. The allocation of tolerance has an effect on manufacturing cost and product quality. The main purpose of tolerance allocation is to manufacture products with least machining cost possible, while meeting all functional requirements of the product. The tolerance chart is an efficient and a sophisticated technique used in industries to

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equating unknown working dimensions to the blue print dimensions.

In this paper, a tolerance chain identification system based on arithmetic approach is established to identify tolerance chains in a process plan. It is also used to formulate the working dimension and their tolerances constraint equations. A nonlinear optimization model is formulated with the machining cost as the objective function and blue print tolerances, blue print dimensions and process capability as constraints. The nonlinear, constraint tolerance allocation problem has been solved using non-conventional optimization technique namely simple Genetic Algorithm (GA).

Ji [7], Ngoi and Kuan [8] and Zhang et al. [9], al presented various techniques for optimization. In order to get the optimized working dimensions and tolerances they have considered different parameters and objectives. Ngoi and Tan [10] developed a „black-box‟ approach for the generation of constraint equations. Here, the process links could be inspected by tagging each feature plane with a unique and meaningful ID. The linear equations were formed and solved by using linear programming methodology. A „window‟ approach was later developed by Ngoi and Seow [11] to locate the process links. The users identify the process links from the model which was constructed from the process plan. Salamons et al. [12] presented a functional tolerance specification to support the user in a semi-automatic manner. A set of equations are generated for the calculations of tolerance analysis. A stepper approach was introduced by Ngoi and Goh [13] to find out the path of the process link. Ngoi and Tan [14] developed a “Maze chart” to trace the process link. Then, the constraints on working dimensions and tolerance are derived from the link. The two sets of linear equations are created and solved by using the goal-seeking linear programming methodology. Jami et al.[15] presented an application of a geometric dimension and tolerance model for use in both design and process planning. The model is based on relative degrees of freedom of geometric entities. Dimension graphs are created based on the degrees of freedom for each control direction. Yeo et al. [16] proposed a method to determine the optimal sequence for production and its optimal process tolerance to attain the minimum production costs.

III. PROBLEM DESCRIPTION The allocation of tolerances in a machining process planning affects the product quality and its efficiency. The machining tolerance range varies with the change in the nominal machining dimensions. The optimum machining tolerance allocation through tolerance charting includes the determination of working dimensions and their tolerances at the lowest machining cost by satisfying the blueprint specifications. One of the main steps involved in the tolerance charting is tolerance chain identification. The tolerance chains are inherent and hidden among the machining cut and the blue print objects. It is used to formulate the tolerance constraint equations through working dimensions constraint equations and also to find both working dimensions and tolerances in machining process. The correct identification of tolerance chain leads to the proper allocation of machining tolerance. The method proposed in this paper ensures the complete and correct determination of tolerance chain in the tolerance chart. In this paper, the main objective is to determine optimum machining tolerances through tolerance charting. The optimality is achieved through the minimization of the machining cost.

Ping-Hung Liu and Chiu-Chi Wei [17] presented a nonlinear programming model to design process tolerances. It is done by minimizing the manufacturing loss due to nonconforming production. Li et al. [18] proposed a methodology for the simultaneous optimization of datum selection and machining tolerance allocation using genetic algorithm. Britton and Thimm [19] developed a new matrix method for determining the working dimensions and offsets for tolerance charts. Shen. Z et Al [20] proposed a procedure for extracting dimension and tolerance stacks for any user defined analysis dimension and automatic part arrangement in assemblies. Sivakumar et al. [21] have applied Differential Evolution (DE) and Non-dominated Sorting Genetic Algorithm (NSGA-II) for tolerance design for minimizing the manufacturing cost. Pérez R and al. [22] proposed an approach for the evaluation process to concurrent conceptual tolerance synthesis in a mechanical design .The reference model was developed and refined through a set of mechanical design study cases. Although the above existing methods are capable of solving the problems on tolerance charting, they are somewhat complex and difficult to solve. Moreover, the previous tolerance chain identification systems are not developed mathematically.

IV. SOLUTION METHODOLOGY The fig. 1 shows the work piece to illustrate the proposed method. Table 1 summarizes the process plan of the given work piece. The reference surface (RS) and processed surface (PS) of the different machining operations are tabulated in table 1. Their working dimensions and working tolerances are also assigned in the table 1. The dimension which measures the distance between the locating surface and the surface being machined is termed as working dimension. The different notations are used to refer the same dimension and tolerance because it denotes the different machining processes. Two dimensional drawing of upper half portion of the work piece is shown in fig.2. The tolerance chart of the work piece is shown in fig.3. The dimension and tolerance is linked with one operation because the working dimensions should be associated with working tolerance. A blueprint dimension is the designed dimension that issued by design engineers. It is the goal of the whole manufacturing process. It has a mean value, sometime called nominal value and a tolerance value. The blueprint dimensions with their blue print tolerances are shown in table 2. The reference and processed surfaces of the blue

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print dimensions are also mentioned in table 2. Table 3 shows the stock removal dimensions with their tolerances.

Fig. 1 Work piece to be Machined

Table 1. Process plan for the component to be machined

Fig. 2. 2D Drawing of upper half of the work piece

Process/ Operatio n Rough Boring Rough Boring Rough Turning Rough Turning Finish Boring Finish Boring Finish Turning Finish Turning Grinding Grinding

A

Working Dimensio n X1

Desired Toleranc e t1

A

C

X2

t2

Lathe 2

A

D

X3

t3

Lathe 2

A

B

X4

t4

Lathe 3

D

A

X5

t5

Lathe 3

A

C

X6

t6

Lathe 4

A

D

X7

t7

Lathe 4

A

B

X8

t8

Grinder 1

A

C

X9

t9

Grinder 2

C

B

X10

t10

Machine

RS

PS

Lathe 1

D

Lathe 1

Table 2 Blue Print Dimensions with tolerances Blue print Surfaces

Dimensions

Tolerance

B-C A-B A-D

Y1=50.0 Y2=10.0 Y3=70.0

0.025 0.075 0.050

Table 3 Stock Removal Dimensions with tolerances Operation Number 0010

Fig. 3 Tolerance chart of the work piece

This section explains the tolerance chain identification system through tolerance charting for the given component as shown in fig. 1. The flow chart of this system is shown in fig. 4. The steps involved in tolerance chain identification system are as follows:

Stock Removal Surfaces

Dimensions

D-A

Tolerance

Solid

0020

A-C

0030 0040 0050 0060 0070 0080

A-D A-B D-A A-C A-D A-B

Solid

Y5 Y6 Y7 Y8

= 1.0 = 1.0 = 1.0 = 0.6

0.250 0.250 0.150 0.400

0090 0100

A-C C-B

Y9 = 0.3 Y10 = 0.3

0.150 0.125

Y4 =3.0

0.300 Solid

The datum (surface) distance Z from the origin of the coordinate system is denoted as DS (Z). A dimension is represented by the equation as follows

Step 1: Set up a coordinate system for the work piece to be machined.

DIM DS(R) - DS (L)

The coordinate system for the work piece to be machined is located in a tolerance chart, as shown in fig. 3. Surface A in the product‟s blueprint is set as an origin of the coordinate system. X‟s represent the working dimensions, the blueprint dimensions and stock removals are represented by Y‟s.

DIM represents dimension (Xi or Yi ), i=1, 2,3..n R represents right surface of the dimension L represents left surface of the dimension DS is Datum (Surface)

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If DS(R) is the right surface a blueprint dimensions Yi and if it is known,. DS(L)  DS(R) - Yi For the example part, DS(C)  Y1  DS (B)  Y1  Y2 (2) Repeat the above step (2) until all distances have been obtained: The flowchart (fig.5) shows the algorithm to find out the distance of all blueprint surfaces.

Fig. 5 Distances of all blue print surfaces

The general table (table 4) can be used to summarize the above result. Table 4 Distances of all blue print surfaces Surfaces A B C D

Positions of the surfaces 0 Y2 Y1+Y2 Y3

Step 4: Find out the direction of each blueprint surface. To obtain the tolerance chains, it is necessary to find out the direction of each blueprint surface. (1) Positive (+) for the surface on the positive direction of the x-axis. (2) Negative (-) for the surface opposite to the negative direction of the x-axis. For surfaces A and C, the direction is negative (-). For surfaces B and D, the direction is positive (+).

Fig. 4 Flow chart for Tolerance chain identification system

Step 2: Find out the surfaces (datums) that are involved in the tolerance chart by analyzing the blue print drawing of the component. It is found to be as four namely A, B, C and D.

Step5: Determine the initial positions of all surfaces of the component from the blue print dimensions.

Step 3: Find out the distance of all blueprint surfaces. An algorithm is developed to find out the distance of all blueprint surfaces (1) For Blueprint dimensions: Surface A is the left surface of a blueprint dimensions Yi i,e DS(R) = Yi., where R is the right surface of dimension Yi. Surface A is the origin of the coordinate system in the blueprint, that is: DS (A) = 0 is a default DS (B) = Y2 and DS (D) = Y3 (2)Remaining blueprint dimensions: DS (L) is the left surface of a blueprint dimensions Yi and if it is known, DS(R)  Yi  S(L)

DS(C) - DS (B)  Y1, DS(B) - DS(A)  Y2 , ………………………..(2) DS (D) - DS (A)  Y3 , DS (A)  0.

The initial positions of all surfaces of the component is DS(A)  0, DS(B)  Y2 , DS(C)  Y1  Y2 , DS (D)  Y3.

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Y1  X10

Step 6: Update the positions of the surfaces in each operation. The position of the surfaces in each operation has been updated step-by-step starting from the last machining cut of the component. (1) The position of the working dimension Xi is obtained from equation (1). (2) The distance table i,e the positions of the surfaces is updated as follows: If M is the machined surface for Xi , and Yk its stock removal , then the distance of the surface M will be updated by adding (+ ) or subtracting (- ) Yk with respect to the direction of surface M. (3) Repeat the above steps from the last working dimension Xn .i,e from X10 to X1 The processed surfaces only change their positions as shown in the table 6. In each machining cut, the amount of stock removal is just added to or subtracted from the processed surface. The evolution of surface changes of all machining-cut objects is shown in table 5.

Y2  X 9 - X10 Y3  X 7 Y4  X1 - X 3 Y5  X 3 - X 5 Y6  -X2  X 3 - X 5  X 6 Y7  X 5 - X 7 Y8  - X 3  X 4  X 5 - X8 Y9  X 9 - X 6 Y10  X8 - X 9  X10

With these tolerance chains, the resultant dimensions can be calculated to check whether the component can be made into the required product when the blue print and stock removal dimensions are all known. Step 9: Formulation of the tolerance constraint equations. The tolerance chains obtained here are very useful for tolerance allocation. The tolerance constraint equation is then formulated from the blueprint dimension equation. For blue print dimension Y1: Y1  X10 and tolerance  0.025 (refer table 2)

Step 7: Determine the working dimensions in terms of blueprint dimensions. The working dimension is the absolute value of the distance between the processed surface and reference surface. In order to determine the absolute value, the reference surface is subtracted from processed surface and ensures that the working dimension value is positive regardless of whether the surface is processed or reference. From table 1, Reference surface: D Processed surface: A D-A= Y3 + Y4 + Y7 - (-Y5) = Y3  Y4  Y7  Y5

Therefore, t10  0.025 Similarly for all other blue print dimensions (table 2) and stock removal dimensions (table 3) are as follows: t 9  t10  0.075 t 7  0.050 t1  t 3  0.300

Similarly for all other working dimensions are as follows

t 3  t 5  0.250 t 2  t 3  t 5  t 6  0.250

X 2  Y1  Y2  Y5 - Y6 - Y9

t 5  t 7  0.150

X 3  Y3  Y5  Y7 X 4  Y2  Y5  Y8  Y10

t 3  t 4  t 5  t 8  0.400 t9  t6

X 5  Y3  Y7

 0.150

t 8  t 9  t10  0.125

X 6  Y1  Y2 - Y9 X 7  Y3 X8  Y2  Y10 X 9  Y1  Y2

V. OPTIMIZATION MODEL The optimal allocation of machining tolerance and economical machining cost of the component are achieved by developing an optimization model. The model involves the formulation of the objective functions and constraints on working dimensions and tolerances. The saving of cost is one of the main objectives of the tolerance chart. So, it is necessary to establish a relationship between cost and tolerance for assigning the tolerance.

X10  Y1

Step 8: Determine the required blue print and stock removal dimensions. The required blue print and stock removal dimensions determined from the working dimensions represents a tolerance chain. It can be determined by applying Gaussian elimination method to the working dimension equations. Obviously, if a different operation sequence is applied to the component, the output tolerance chains must be different.

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Table 5 Evolution of surface changes in the tolerance chain Positions of the surfaces(Datums) Datum Surfaces Direction

A

B

C

D

(-)

(+)

(-)

(+)

Working Dimensions

Reference Surface

Processed Surface

X10

C

B

0

Y2

Y1+Y2

Y3

X9

A

C

0

Y2+Y10

Y1+Y2

Y3

X8

A

B

0

Y2+Y10

Y 1 + Y2 - Y9

Y3

X7

A

D

0

Y2 + Y10 + Y8

Y 1 + Y2 - Y9

Y3

X6

A

C

0

Y2 + Y10 + Y8

Y1+Y2-Y9

Y3 + Y7

X5

D

A

0

Y2 + Y10 + Y8

Y1 + Y 2 - Y9 - Y6

Y3+Y7

X4

A

B

-Y5

Y2+Y8+Y10

Y1 + Y 2 - Y9 - Y6

Y3+Y7

X3

A

D

-Y5

Y2+Y8+Y10+Y4

Y1 + Y 2 - Y9 - Y6

Y3+Y7

X2

A

C

-Y5

Y2+Y8+Y10+Y4

Y1 + Y2 - Y6 - Y9

Y3 + Y7 + Y 4

X1

D

A

-Y5

Y2+Y8+Y10+Y4

Y1 + Y 2 - Y6 - Y9

Y3 + Y4 + Y 7

Table 6 Optimization model Parameters & Objective Function

Constraints

X10  50

Working dimensions

X 9 - X10  10 X 7  70

n

Max 

X

Process Capability Constraints

i

X1 - X 3  3.0

i 1

X 3 - X 5  1.0 - X 2  X 3 - X 5  X 6  1.0

Not Applicable

X 5 - X 7  1.0 - X 3  X 4  X 5 - X8  0.6 X 9 - X 6  0.3 X8 - X 9  X10  0.3 Working tolerances n

Min.Cost 

 ( A  tB ) i 1

k i

t10  0.025 t 9  t10  0.075 t 7  0.050 t1  t 3  0.300 t 3  t 5  0.250 t 2  t 3  t 5  t 6  0.250 t 5  t 7  0.150 t 3  t 4  t 5  t 8  0.400 t9  t6

 0.150

t 8  t 9  t10  0.125

Where Xi is the working dimension for the ith operation A, B, k are the coefficients of the cost tolerance relationship , ti is the working tolerance for the ith operation and n is the number of operations.

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0.0020  t1  0.0080 0.0015  t 2  0.0060 0.0020  t 3  0.0080 0.0008  t 4  0.0030 0.0015  t 5  0.0060 0.0015  t 6  0.0060 0.0015  t 7  0.0060 0.0008  t 8  0.0030 0.0004  t 9  0.0010 0.0004  t10  0.0010


The optimal tolerance allocation can be achieved through tolerance charting. Maximization of working dimensions and minimization of total machining cost (summation of the machining cost involved in the machining of all the tolerance dimensions) are the objective functions involved in the optimization model. Kenneth W. Chase, 1999 established the cost functions based on experimental data. The mathematical model based on the machining cost is formulated. Table 6 shows the optimization model developed. Table 7 shows the cost values of various coefficients involved in the cost function [23]. Table 7 Cost Table Process/ Operation Rough Boring Rough Boring Rough Turning Rough Turning Finish Boring Finish Boring Finish Turning Finish Turning Grinding Grinding

A 1 1 1 1 1 1 1 1 1 1

Values of Coefficients B K 0.11804756 0.45747142 0.11800302 0.4389869 0.11804756 0.45747142 0.07201641 0.46822793 0.11800302 0.4389869 0.11800302 0.4389869 0.11800302 0.4389869 0.07201641 0.46822793 0.0133561 0.7827624 0.0133561 0.7827624

Fig. 6 Flow Chart representing the optimization process using the proposed GA [21]

VII. RESULTS AND DISCUSSION Tolerance charting is a graphical tool which is used to calculate the working dimensions and their tolerances for manufacturing a product. To control the tolerance buildups, tolerance chart is developed. An important step in the tolerance charting process is tolerance chain identification. The main use of the tolerance chain identification is to calculate the working dimension and tolerance of each operation. The tolerance chain model has improved the capability of the tolerance chart. Over the existing methods, this proposed method is developed to solve the difficulties such as time consuming, tedious that are involved in the tolerance chain identification. In contrast with the earlier work, this method has several benefits:

Thus a nonlinear, constraint tolerance allocation problem is developed. This problem has been solved by using non-conventional optimization technique namely simple Genetic Algorithm (GA) VI. GENETIC ALGORITHM (GA) In principle, GA is search algorithm based on mechanics of natural selection and natural genetics. They combine survival of the fittest among the string structures with randomized yet structured information exchange to form a search algorithm with innovative flair of natural evolution. A GA starts with a random creation of a population of strings and thereafter generates successive population of strings that improve over time. The process involved in the generation of new populations mainly consists of the operations such as reproduction, crossover and mutation. Flowchart in fig.6 shows the process that the proposed GA takes to find an optimum solution. Table 8 shows the parameters involved in GA.

(i) (ii) (iii)

(iv) (v)

Table 8 Parameters involved in GA Parameters Number of objective function Objective function Crossover Type Strategy : Population size Total no. of generations Cross over probability Mutation probability String length Number of variables (Binary) Sigma-share value Sharing Strategy

Conditions /Values 1 Minimize Binary GA (Single-Pt) 1 cross - site with swapping 100 100 0.8 0.1 100 10 0.3970 Parameter Space

(vi)

Quicker knowledge gaining time for the new user The method is easy to learn and to implement. Time required for generating the working dimensions and working tolerance equations is less. The proposed method reduces the purpose of applying complicated matrices or arrays. Creation of detailed graphical aids is not required to formulate the constraints on working dimensions and working tolerances. The calculation of balance dimensions is eliminated.

The optimal machining tolerances are allocated through the appropriate selection of tolerance charting model. A reciprocal power nonlinear optimization model was developed with the machining cost as the objective function and blueprint tolerances, process capabilities as the constraints. A GA based optimization procedure was developed to arrive the optimal machining tolerance of each

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in the given machine component. Table 9 shows the optimum working dimensions and the working tolerances with minimum machining cost obtained using GA. From the results, it is observed that the machining cost for the finish turning process (operation number 0070 and 0080) is higher than other machining process. Fig. 7 shows the GA result history.

process engineers may want to relax an unnecessarily tight working tolerance.

References [1]

[2]

Table 9 Optimum Parameters obtained using GA Working Dimensions X1 2.95

Working Tolerances t1 0.00676

Machining Cost (in dollars) C1 2.160899

X2

2.35

t2

0.00534

C2

2.134398

X3

2.83

t3

0.00711

C3

2.173472

X4

0.47

t4

0.00259

C4

2.138589

X5

2.80

t5

0.00572

C5

2.143872

X6

2.35

t6

0.00566

C6

2.817971

X7

2.76

t7

0.00197

C7

4.206474

X8

0.41

t8

0.00250

C8

4.206474

X9

2.36

t9

0.00091

C9

2.171108

X10

1.96

t10

0.00091

C10

2.190662

C

26.342

Total Machining Cost

[3]

[4]

[5]

[6] [7]

[8]

GA Result History

[9]

Cost (in dollars)

29 28

[10] 27 26

[11]

25 1

10

19

28

37

46

55

64

73

82

91 100

[12]

Generation Num ber

Fig. 7 GA Result History [13]

VIII. CONCLUSION During machining, it is necessary to control the work dimensions of the component to be machined. Tolerance chart is a graphical representation tool which shows the sequence of machining operations on a work piece. It also shows the working dimension, the working tolerance and the amount of stock to be removed at each step in the sequence. In this paper, tolerance chain identification system based on arithmetic approach is developed and is used to generate the constraints on blue print dimensions, tolerances and intermediate machining stock removals in the optimization process.

[14]

The tolerance chain identification plays a crucial role in the tolerance chart throughout the whole process. As to the nature of the machining cuts, only position changes of the machined surfaces in the coordinate system occur during the machining process. After the tolerance chains are identified, the remaining work to be done in the tolerance chart is simplified. When analyzing a tolerance chart,

[18]

[15]

[16]

[17]

[19]

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