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Yuan Mao Huang

Professor e-mail: ymhuang@ntu.edu.tw

Ching-Shin Shiau

Research Assistant Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan, Republic of China

Optimal Tolerance Allocation

for a Sliding Vane Compressor

An optimization model has been built with consideration of the required reliability, the minimum machining cost, and quality loss. The normal and the lognormal distributions of tolerances that depend on the production types of components are used in the reliability model. Cost tolerance data obtained from Bjørke are used to calculate the machining cost. The asymmetric quadratic quality loss model is used to calculate quality loss caused by the deviation and the mean-shift of distributions. Tolerance allocation of a sliding vane rotary compressor is optimized for the required reliability, the minimum cost and quality loss, and optimum tolerances of components are recommended. The results show that high accuracies of the slot length, the vane thickness, and the slot width are required. Hence, their tolerances are smaller than other components. The effects of the correlation coefficient of the bottom cover plate and the top cover plate and the correlation coeffi-cient of the front cover plate and the rear cover plate to total cost are insignificant. Further, the cost of quality loss is reduced when the weighting ratio of the quality loss function weighting coefficient to the machining cost function weighting coefficient is increased. The total cost is increased because tight tolerance allocation increases the machining cost. 关DOI: 10.1115/1.2114893兴

Introduction

Sliding vane rotary compressors have been developed and used for years because of lightweight, small size, simple mechanism, and easy maintenance. They are commonly used in refrigerators and automotive air conditioning systems. A considerable amount of studies were conducted to improve sliding vane rotary com-pressors关1兴.

A sliding vane rotary compressor has been designed and its schematic drawing is shown in Fig. 1关2兴. Some topics regarding this compressor were studied, such as the static contact force, stress and deflection of its vanes, stator contour, impact of vanes on the rotor, configuration, disassembly process, and test of the compressor关2–5兴. The results showed that the compression ratio was not high as expected and the noise level was high because of leakage and friction due to improper tolerance.

Tolerance design is an important task in embodiment design. Tolerance allocation of an assembly in the conventional design is based on designers’ experience or design handbooks. When toler-ance is tight, the cost is high, and vice versa. If accumulated tolerance of components is out of specification and causes im-proper assembly, tolerances of components have to be revised. The task becomes complicated and very tedious if the number of components is large. Furthermore, the highest reliability, the best quality, and the minimum cost of a product cannot be assured and achieved.

The standard for the exchange of product model data共STEP兲 proposed by the international standard organization共ISO兲 and the product data exchange specification共PDES兲 proposed by the US National Bureau of Standards were very useful for tolerance de-sign. Bjørke关6兴 and Juster et al. 关7兴 studied the relationship of linear tolerance accumulation and allocation. Feng and Yang关8兴 generated a model to integrate tolerance information into the product database. Parkinson关9兴 applied a deterministic method of robust design to determine optimum nominal dimensions of com-ponents with dimensional tolerances. Zhou et al.关10兴 and Jordaan and Ungerer 关11兴 used various numerical methods to approach optimal tolerance allocation. Di Stefano关12兴 analyzed statistical tolerance by using a mean shift model to evaluate the mean shift

factor. Hong and Chang关13兴 deal with interactions of the devia-tion space, deviadevia-tion volume, and tolerance primitives to deter-mine the relationship between process planning decisions and tol-erances. Main and Ferreira关14兴 addressed the acceptable level of inaccuracy of the machined parts associated with its geometric tolerances. Manarvi and Juster关15兴 compiled available informa-tion and developed an integrated tolerance model and a tolerance allocation process. Hu et al.关16兴 provided automated methods for the specification of geometric tolerance types. Wu and Rao关17兴 investigated the modeling and analysis of tolerances and clear-ances. Mujezinovi’c et al.关18兴 applied a mathematical model for representing geometric tolerances to polygonal faces and showed its sensitivity to the precedence of datum reference frames.

Dong et al.关19兴 studied the relationship between tolerance and the machining cost. Shiu et al.关20兴 discussed misalignment and fabrication error of compliant parts and minimized manufacturing costs associated with tolerances. Xue and Ji关21兴 investigated a cost-tolerance model and allocated systematically the best toler-ance to minimize the total manufacturing cost.

Mahadevan and Ni 关22兴 analyzed tolerance reliability assess-ment of automotive spot-welded joints. Several researchers imple-mented the quality loss model关23–26兴 into tolerance synthesis. Ye and Salustri关27兴 constructed a nonlinear optimization model with consideration of product design, manufacturing, and quality si-multaneously. Li关28兴 investigated the optimal manufacturing set-ting to minimize the expected quality loss.

Up to the present, only the combined cost tolerance and reli-ability model 关29,30兴 and the combined cost tolerance and the quality loss model关23–25,31,32兴 were utilized for tolerance allo-cation. The optimal tolerance allocation model is not available based on the simultaneous consideration of the reliability, cost, and quality loss. In addition, semi-tolerances were assumed in existing studies. Therefore, the purpose of this study is to build an optimal tolerance allocation model with simultaneous consider-ation of the required reliability, the minimum cost, and quality loss. In the meantime, since tolerance allocation of the compressor as shown in Fig. 1 has not been optimized, tolerances between vanes and slots in the rotor, and vanes and the stator will be optimized.

Contributed by Design for Manufacturing Committee of ASME for publication in the JOURNAL OFMECHANICALDESIGN. Manuscript received May 1, 2004; final manu-script received May 1, 2005. Assoc. Editor: David Kazmer.

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Method of Approach

Basic components of the compressor as shown in Fig. 1 consist of a rotor, a stator, and sliding vanes. A rotor with ten slots for ten sliding vanes is inside and concentric with the stator. The outer surface of the rotor, the inner contour of the stator and the vanes form a volume segment. When the rotor rotates, the vanes slide outward along the slot due to the centrifugal force. The maximum volume segment is obtained at the end of the intake process. When the vanes slide inward, the volume segment is reduced and the air is compressed. The cross section of the compressor is shown in Fig. 2 and 18 independent components are 1—stator, 2—rotor, 3—bottom cover plate, 4—top cover plate, 5—front cover plate, 6—rear cover plate, 7—vane, 8—front bearing, 9—rear bearing, 10—oil ring, 11—bottom bolt, 12—top bolt, 13—front bolt, 14—rear bolt, 15—bottom gasket, 16—top gasket, 17—front gasket, and 18—rear gasket.

The reliability index is the minimum distance from the origin of the coordinate system to the limited surface of the design. The standard parts of an assembly that are manufactured in mass pro-duction are considered as the normal distribution. The non-standard parts of an assembly that are produced in batch produc-tion are considered as the lognormal distribuproduc-tion with a dimension mean-shift because of the asymmetric distribution关33兴.

Let X¯ be the mean vector of the normal distribution vector or the lognormal distribution X, V be the covariance matrix, and cov共xi, xj兲 stand for the covariance of the variable dimensions xi and xjin an assembly关34兴 as X =兵x1,x2, . . . ,xn其 共1兲 X ¯ = 兵x¯1,x¯2, . . . ,x¯n其 共2兲 V =

␴1 cov共x1,x2兲 ¯ cov共x1,xn兲 cov共x2,x1兲 ␴2 ¯ cov共x2,xn兲 ] ]  ] cov共xn,x1兲 cov共xn,x2兲 ¯ ␴n

共3兲 where cov共xi,xj兲 = E关共xi−␮i兲共xj−␮j兲兴 共4兲

iand␮jare the mean values of xiand xj, respectively, and␴iis

the standard deviation of xi. The coefficient of variation␦iof xiis

i=

i

i

共5兲 The variables of the normalized standard distribution for the nor-mal distribution or the lognornor-mal distribution, respectively, are

yi= xi−␮ii 共6兲 and yi= ln xi−␮ii 共7兲 The variable vector of the normalized standard distribution is Y =兵y1, y2, . . . , yn其. The correlation coefficient of yiand yjis

ij=

1 N

i=1

N

yiyj 共8兲

where N is the population size. The independent standard normal distribution vector Z =兵z1, z2, z3, . . . . , zn其 of variables zican be ob-tained from y1=␣11z1 y2=␣21z1+␣22z2 ¯ yn=␣n1z1+␣n2z2+¯ + ␣nnzn 共9兲 where ␣11= 1.0 ␣i1=␳ij 1⬍ iik= 1 ␣kk

ij

j=1 k−1ijkj

1⬍ k ⬍ iii=

1 −

j=1 i−1ij2 共10兲

Then the variables ziare

z1= y1 z2= 1 ␣22 共y2−␣21z1兲 . . . zn= 1 ␣nn 共yn−␣n1z1. . . −␣n,n−1zn−1兲 共11兲

Once the independent normal distribution vector Z is available, the reliability index can be obtained by using the Beta algorithm which was developed by Hohenbichler and Rackwitz 关35兴 as shown in Fig. 3, where G is the design function. Details of the Beta algorithm are presented in the Appendix.

Fig. 1 Components and schematic drawing of sliding vane ro-tary compressor

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In the case of multiple design functions, the minimum reliabil-ity index is

␤min= min兵␤1,␤2, . . . ,␤k, . . . ,␤p其 1 艋 k 艋 p 共12兲

where p is the number of design functions.

The quality loss model of the-nominal-the-best type is that the best value of dimension is equal to the nominal dimension. If it is used, the expected total loss is关36兴

E共Lasm兲 =

i=1 n

i⬍mi kiL关␴i2+共␮i− mi兲2兴 +

i⬎mi kiR关␴i2+共␮i− mi兲2兴

共13兲 where mistands for the nominal value of the ith dimension

kiL= AhiL 2 共14兲 kiR= AhiR2

are quality loss coefficients, Ahis the cost of the replacement or the repair for the hth component, andiL and ⌬iR are specific

semi-tolerance, or called customer’s semi-tolerance of the ith di-mension. The quality loss curve and the probabilistic density func-tion are shown in Fig. 4.

It can be seen that quality loss consists of loss by the deviation, kiLi2 and kiR␴2i, and loss by the mean-shift, kiL共␮i− mi兲2 and kiR共␮i− mi兲2. Quality loss increases when the standard deviation

increases or the mean-shift increases.

Since the reliability index stands for the minimum distance from the origin of the coordinates to the limited surface in the standard normal distribution, the reliability corresponding to the reliability index can be obtained from the inverse of the accumu-lated distribution function based on the specified reliability. The left and right side semi-tolerances provide the required informa-tion related to quality loss. Söderberg关23兴 defined that the cus-tomer’s lower acceptance limit of a component life is 75% of the nominal value. The reliability index of 75% reliability is

␤*=−1共0.75兲 = 0.67449 共15兲

where⌽ is the accumulated functional distribution of the inde-pendent normal distribution. Customer’s semi-tolerance can be optimized for minimizing the goal of the machining cost through the machining cost-tolerance model Cm共⌫兲 with the constraint

␤*

k共⌫兲 艋 0 1 艋 k 艋 p 共16兲

where

⌫ = 兵␴1,␴2, . . . ,␴i, . . . ,␴n其 1 艋 i 艋 n 共17兲

The process of calculating the quality loss coefficient is shown in Fig. 5.

If the replacement cost Ahis known, a set of⌬iLand⌬iRnearest

to the specific customer’s quality loss can be obtained from Fig. 4. In the meantime, the quality loss coefficients kiL and kiR of the asymmetric quality loss function can be obtained. Then tolerance can be optimized by a non-linear programming with minimizing the goal of

Ct共⌫兲 = WmCm共⌫兲 + WqE共Lasm共⌫兲兲 共18兲

and constraint function ␤**

k共⌫兲 艋 0 1 艋 k 艋 p 共19兲

where Ctis the total cost that consists of the machining cost Cm

and the quality loss expected value E共Lasm兲, Wmand Wqare the weighting coefficients of the machining cost function and the quality loss function, respectively, ␤** is the required assembly reliability index, and p is the number of design functions. The reliability index␤kof each design function stands for the assem-bly yield rate of each component. If␤**is larger, the reliability of

the assembly will be higher. When the reliability is set at 99.0%, the corresponding reliability index is 2.32635. The processes to obtain semi-tolerance are shown in Fig. 6.

The minimization of machining cost and quality loss is a multi-attribute decision-making problem. Also, the desired reliability

in-Fig. 3 Flowchart of beta algorithm process

Fig. 4 Asymmetric quality loss curve and asymmetric dimen-sion distribution

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dex can be set in a penalty function form as the third objective function. Due to the consideration of computational efficiency, the attribute functions can be added together with different weighting coefficients. Cheng and Maghsoodloo关26兴 directly summed the cost-tolerance and quality loss for tolerance allocation, but they only considered the normal distribution condition without dimen-sional correlation. In this study, the replacement cost in the quality loss function has been carefully calculated in order to have an equivalent unit as machining cost in Bjørke’s cost-tolerance model. Therefore, machining cost function and quality loss func-tion can be combined into a single objective funcfunc-tion with se-lected weighting coefficients. The weighting ratio Wq/ Wm repre-sents the relative importance between quality loss and machining cost.

Oriented Functional Relationship. If an assembly that

con-sists of many components is complicated, it is not worth analyzing all dimension features because of the cost. The assembly feature graph or the functional feature graph can be used to determine key functional dimensions. The representation of the oriented func-tional relation for dimension tolerance proposed by Zhang and Porchet 关37兴 is used in this study, and its main purpose is to organize features of components according to assembly drawings. Users can find functional components, which really affect the per-formance of a product, quickly from the data bank and analyze tolerance features.

The oriented functional relationship is shown in Fig. 7. Each node stands for an independent component, and a line with an arrow stands for the mating relation between components. If an arrow points from A to B, it stands for component B is fixed on component A. A line without an arrow stands for a sub-assembly with components as points C and D, and these components are assembled into a sub-assembly before being assembled with other components. A dashed line stands for the functional dimension that needs to be controlled for the dimension and tolerance, as point E to point F.

Six symbols to represent relations between components are shown in Table 1. The connection between planes and the connec-tion between cylinders stand for a tight fitting between two planes and two cylinders, respectively. The connection by planes and planes and the connection by cylinders and cylinders stand for a distance existed and loose running fit for planes and cylinders, respectively. The connection by cylinders and planes stands for simultaneous existence of connection between planes and

connec-tion between cylinders.

The oriented functional relationship graph of the compressor as shown in Figs. 1 and 2 is shown in Fig. 8. The bottom cover plate that is used as the basis is at the top. Only a line contact, but without the assembly relationship, exists between vanes and the stator inner contour. The rotor connects to the inner rings of two bearings. Each bearing is considered as a sub-assembly that con-sists of the outer ring and the inner ring with the relative motion between them and cannot be counted as one node. Hence, nodes 8 and 9 are divided into 81and 82, and 91and 92, respectively. The

rotor and vanes are considered as the third sub-assembly. It is necessary to find tolerance chains of functional dimensions in order to analyze dimension tolerance. There are four dashed lines. If a functional dimension chain consists of these specific dimensions, all tolerances in the dimension chain should be ana-lyzed. The dashed line from node 1 to node 7 stands for the contact between vanes and the stator inner contour. When the rotor is stationary, vanes may not contact with the stator. When the rotor rotates, vanes slide outward due to the centrifugal force. The tip of the vane slides along the stator inner contour. Although the dashed line represents the special contact situation in the oriented functional relationship graph, there actually is no functional toler-ance between vanes and the stator. Therefore, it is not necessary to analyze this dimension chain.

Honda et al. 关38兴 utilized a numerical method to analyze the gap flow between the side cover plate and the rotor and proved the importance of the clearance to the compressor performance. Two dashed lines from node 2 to nodes 5 and 6 stand for functional dimensions of the rotor to the front cover plate and the rear cover plate, respectively. These two clearances are very important to the compressor. When the clearance is decreased, friction between the rotor and two cover plates, power loss and wear of components increase. If the clearance is increased, leakage of fluid occurs and reduces the compression performance. Therefore, all the

dimen-Fig. 6 Flowchart of determining optimal semi-tolerance

Fig. 7 Oriented functional relationship

Table 1 Symbols for feature fit in oriented functional relationship

Relation of connection Symbol

Connection between planes p

Connection between cylinders c

Connection by planes and planes pp

Connection by cylinders and cylinders cc

Connection by cylinders and planes cp

Thread connection t

Fig. 8 Oriented functional relationship graph of rotary compressor

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sion chains related to these two clearances need to be controlled. The first functional dimension chain consists of six nodes as shown in Fig. 9共a兲. The dimension variables are shown in Fig. 9共b兲. The length of slot in the rotor, x2, plus two specific clear-ances C1and C2should equal the stator length x1plus the

thick-ness of two gaskets L1and L2. The constraint is

x1+ L1+ L2= x2+ C1+ C2 共20兲 There are four nodes in the second functional dimension chain as shown in Fig. 10共a兲. Nodes 91and 92stand for the outer ring and the inner ring of the bearing. The thickness of the bearing plus the wall thickness of the slot L3should equal the thickness of the

real cover plate x4. The locations of dimension variables are shown in Fig. 10共b兲. The constraint can be written as

x3+ L3+ C1= x4+ C1 共21兲 The third functional dimension chain as shown in Fig. 11 is similar to the second functional dimension chain, except with one more oil ring, node 10. The bearing thickness L4, the oil ring

thickness plus the wall thickness x5should be equal to the front cover plate thickness x6as shown in Fig. 11共b兲. The constraint can

be written as

C2+ L4+ L5+ x5= C2+ x6 共22兲

In addition, a dashed line that connects nodes 2 and 7 stands for the clearance between vanes and the rotor. The rotor and the vane form a sub-assembly, and the fourth functional dimension chain is shown in Fig. 12共a兲. The schematic drawing of the vane and the rotor slot is shown in Fig. 12共b兲, and the dimension relation is

x7+ C3+ C4= x8 共23兲 Synthesis and Optimization of Tolerance. The design

func-tions as shown in Fig. 13 can be rewritten into

G1=共x1+ L1+ L2兲 − 共x2+ C1+ C2兲 + 0.1 共24兲 G2= x3+ L3− x4+ 0.1 共25兲

Fig. 9 First functional dimension chain

Fig. 10 Second functional dimension chain

Fig. 11 Third functional dimension chain

Fig. 12 Fourth functional dimension chain

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G3= x4− x3− L3+ 0.1 共26兲 G4= L4+ L5+ x5− x6+ 0.1 共27兲 G5= x6− L4− L5− x5+ 0.1 共28兲 G6= x8−共x7+ C3+ C4兲 + 0.02 共29兲

The design function G1stands for the clearance constraint be-tween the front cover plate and the rear cover plate. The sum of clearances is within 0.1 mm. The design functions G2 and G3

mean that the thickness variation between the rear cover plate and the bearing is within 0.1 mm. The design functions G4 and G5

mean that the thickness variation between the front cover plate, the bearing and the oil ring is within 0.1 mm. The purpose of the first five design functions is apparently to maintain the appropriate clearances C1 and C2between two cover plates and bearings in

order to prevent leakage and reduce friction. The design function G6stands for the required clearance between the vanes and the

slots of the rotor to allow vanes to slide smoothly. The combined nominal clearance of C3 and C4 is 0.05 mm and the minimum clearance is 0.02 mm.

The covariance matrix of dimension variables is

V⬘=

1 ␳12 ¯ ␳1n ␳21 1 ¯ ␳2n ⯗ ⯗  ⯗ ␳n1n2 ¯ 1

共30兲 According to Parkinson关9兴, the dimensions on the same compo-nent are correlated. It can be considered that the different dimen-sions at the same component may be manufactured by the same datum setup during the process. Holes for the bearings and the oil ring in the front cover plate and the rear cover plate are manufac-tured by the milling process. The parts are set up on the same fixture when machining for the plate thickness and the holes. The dimension variables x3and x4located at the rear cover plate are dependent, and the dimension variables x5and x6located at the

front cover plate are dependent. Therefore, the values of the cor-relation coefficients␳34and␳56are not zeros. The other

dimen-sion variables are assumed independent and their correlation co-efficients are zero. Equation共30兲 can be rewritten as

V⬘=

1.0 0 0 0 0 0 0 0 0 0 0 0 1.0 0 0 0 0 0 0 0 0 0 0 0 1.0 ␳34 0 0 0 0 0 0 0 0 0 ␳43 1.0 0 0 0 0 0 0 0 0 0 0 0 1.0 ␳56 0 0 0 0 0 0 0 0 0 ␳65 1.0 0 0 0 0 0 0 0 0 0 0 0 1.0 0 0 0 0 0 0 0 0 0 0 0 1.0 0 0 0 0 0 0 0 0 0 0 0 1.0 0 0 0 0 0 0 0 0 0 0 0 1.0 0 0 0 0 0 0 0 0 0 0 0 1.0

共31兲 Optimal tolerance allocation can be achieved by following the process mentioned above.

Results

The combined model is evaluated for optimal tolerance alloca-tion of an assembly with normal distribualloca-tions of independent vari-ables without the mean-shift and quality loss. The results are com-pared and show comparable agreement with the existing data关39兴 to assure the feasibility of the combined methodology.

The statistical distributions of components depend on manufac-turing processes. The oil ring, bearings, and gaskets are standard parts made in mass production, and the dimensions are assumed to be normal distributions. The other components are manufactured in batch production, and the dimensions are assumed to be log-normal distributions as shown in Table 2.

Only the lognormal distributions of dimension variables x1to x8 will be optimized, and these tolerances must be specified in design drawings. Dimension variables L1to L5are from standard parts

made in mass production and will not be optimized. Normal dis-tributions of these dimensions affect the reliability index in the reliability model. Hence, the standard deviations of these dimen-sions must be controlled to have good quality products.

Since costs of some compressor components are not available, the process of this study is based on the cost tolerance model proposed by Bjørke关6兴. The numerical function in the reciprocal power type is used for the curve fitting, and the coefficients of the curve are listed in Table 3. The cost tolerance coefficients of all functional dimensions of components are listed in Table 4. Be-cause L1to L5are the dimensions of the standard parts and C1to C4are specified clearances, they are not included in the optimiza-tion model. If the required reliability is 75% and the required reliability index ␤* is equal to 0.67449, customer’s semi-tolerances of all dimension variables are listed in Table 5.

Based on the manufacturing cost, the replacement costs of the stator A1, the rotor A2, the rear cover plate A3, the front cover plate A4, and the set of vanes A5are 9.0, 5.0, 1.0, 1.5, and 0.3, respec-tively关6兴. Quality loss coefficients of the functional dimensions obtained are listed in Table 6. Because the replacement cost of the rotor is high and the customer’s semi-tolerances of x2and x8are

smaller than other dimensions, the quality loss coefficients of x2 and x8are much larger than other dimension variables. It can be expected that the dimension variables with the required tightest tolerances are x2and x8in tolerance optimization.

Table 2 Statistical distribution types of dimensions

Symbol Name Distribution type

x1 Stator length Lognormal

x2 Rotor center length Lognormal

x3 Rear cover plate sink thickness Lognormal

x4 Rear cover plate thickness Lognormal

x5 Front cover plate sink thickness Lognormal

x6 Front cover plate thickness Lognormal

x7 Vane thickness Lognormal

x8 Slot width Lognormal

L1 Gasket thickness Normal

L2 Gasket thickness Normal

L3 Real cover plate bearing thickness Normal

L4 Front cover plate bearing thickness Normal

L5 Oil ring thickness Normal

C1 Rear cover plate clearance Fixed target value

C2 Front cover plate clearance Fixed target value

C3 Clearance between vane and slot Fixed target value

C4 Clearance between vane and slot Fixed target value

Table 3 Coefficients of curve fitting with Cm= c0t−c1 for

Bjørke’s cost-tolerance models

No. Process type c0 c1

1 External cylindrical surface共rotating兲 1.192 0.591 2 Internal cylindrical surface共rotating兲 7.602 0.814 3 Face surrounding bore, to a datum共rotating兲 18.064 1.115 4 Internal cylindrical surfaces共non-rotating兲 12.435 0.566 5 Face surrounding a bore, to a datum共non-rotating兲 18.622 0.564

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Distribution of Vane Thickness. Two sets of 10 vanes, a total

of 20 vanes, are measured by a digital micrometer to obtain the mean-shift value of the thickness. The thickness of each vane is measured seven times. The largest and the smallest values are obsolete, and the other five values are used to calculate the mean value to reduce the measured error. The mean values of the mea-sured vane thickness are listed in Table 7.

The mean value and the standard deviation of the normal dis-tribution for the vane thickness are 4.948 and 0.0178, respectively. The middle value and the coefficient of variation of the lognormal distribution are 4.947 and 0.0403, respectively. The curves of the probability density function of the normal distribution and the lognormal distribution and the practical distribution histogram are shown in Fig. 14.

The chi-square test is used to verify the fitting goodness of the normal distribution and the lognormal distribution. The results are shown in Table 8. The symbol niis the observed frequency for each interval and ei is the theoretical frequency. From the test

results, fitting of the lognormal distribution is better than that of the normal distribution for the vane thickness.

Since only the dimension variables of the lognormal distribu-tions are used for tolerance optimization in this study, three pa-rameters that are needed for optimization are the mean-shift value, the correlation coefficient of the lognormal distribution, and the standard deviation of dimension variables of the normal distribu-tion. First, it is assumed that the vanes, the front cover plate, and the real cover plate are to be processed by the same machine. Therefore, based on the mean-shift value of the vane thickness x7,

the mean-shift values of the rear cover plate x4and the front cover plate x6 are assumed to be 0.003 mm. It is assumed that other

dimensions do not have the mean-shift. Second, the correlation coefficients of two parts depend on the machining process and the fixture datum setup. In this case, the dimensions of x3and x4, and x5and x6are assumed to be independent. Third, the statistical data of the mass production parts are not available. So, it is assumed that standard deviations of these standard parts are 0.01 mm. Tol-erances of these standard parts are 0.06 mm obtained from the concept of six-sigma, ±3␴. If all of the correlation coefficients are assumed to be zero, the result of tolerance optimization is listed in Table 9. The dimensions with specified tolerances for practical engineering drawings are listed in Table 10. The upper and lower tolerances of functional dimensions are shown in Fig. 15.

The correlation coefficients␳34of x3and x4and␳56of x5and x6

can be set as variables based on optimal tolerances obtained to perform the parameter analysis. Since the machining processes are similar, these two correlation coefficients are set as the same

vari-Table 4 Nominal dimensions and coefficients of cost toler-ance function Cm= c0t−c1

Symbol

Nominal dimension 共mm兲

Coefficients of cost tolerance function

c0 c1 x1 152.0 18.622 0.564 x2 154.8 18.064 1.115 x3 5.0 12.435 0.566 x4 20.0 13.457 0.764 x5 5.0 12.435 0.566 x6 30.0 13.457 0.764 x7 4.95 13.457 0.764 x8 5.0 13.457 0.764 L1 1.5 L2 1.5 L3 15.0 L4 15.0 L5 10.0 C1 0.1 C2 0.1 C3+ C4 0.05

Table 5 Customer’s semi-tolerance of dimension variables

Symbol Left customer’s semi-tolerance⌬iL共mm兲 Right customer’s semi-tolerance⌬iR共mm兲 x1 0.3341 0.3347 x2 0.1303 0.1304 x3 0.2808 0.2945 x4 0.2103 0.2121 x5 0.2801 0.2938 x6 0.2100 0.2112 x7 0.0509 0.0513 x8 0.0507 0.0511

Table 6 Quality loss coefficients of functional dimensions

Symbol

Left quality loss coefficient kiL

Right quality loss coefficient kiR x1 80.6491 80.3548 x2 294.4430 294.0312 x3 12.6799 11.5238 x4 22.6086 22.2152 x5 19.1211 17.3823 x6 34.0203 33.6257 x7 115.9523 113.9805 x8 3500.6748 3441.9239

Table 7 Mean values of measured vane thickness

No Thickness共mm兲 No Thickness共mm兲 No Thickness共mm兲 No Thickness共mm兲

1 4.917 6 4.949 11 4.976 16 4.943

2 4.950 7 4.958 12 4.945 17 4.945

3 4.941 8 4.946 13 4.939 18 4.969

4 4.935 9 4.951 14 4.925 19 4.974

5 4.930 10 4.949 15 4.965 20 4.977

Fig. 14 Probability density functions of normal and lognormal distributions for vane thickness

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able␳. The relation between the correlation coefficient ␳ and the cost is shown in Fig. 16. The second parameter analysis per-formed is to set the standard deviations of the mass production parts as the variables and analyze the relation between the quality of standard parts and the cost. The result is shown in Fig. 17. Increasing the weighting ratio Wq/ Wmincreases the relative

im-portance of quality loss in the combined objective function. The effect of weighting ratio on the cost is studied, and the results are shown in Fig. 18.

Discussion

The required reliability, the minimum machining cost and qual-ity loss are considered simultaneously for tolerance allocation de-sign to improve existing models. Consideration of the required reliability, the minimum machining cost and quality loss simulta-neously is a multi-attribute decision-making problem. In general, the multi-attribute utility analysis is a good way to obtain the optimum results in many applications. Since the manufacturing costs of compressor components are obtained from machining fac-tories in this study, and the replacement cost Ahis converted into the equivalent cost as the same unit as Bjørke’s cost-tolerance model, the gradients of the distribution curves can be obtained. The advantage of single-objective optimization is that the effi-ciency is higher and calculation time is less than the multi-attribute utility analysis. Therefore, the Beta algorithm process and the non-linear programming are used in this study. Hohen-bichiler and Rackwitz’s Beta algorithm is used to calculate the reliability index关35兴 because the efficiency of the transformation is higher than those of other transformation methods when the distribution model is non-normal and asymmetric.

Table 5 shows that right customer’s semi-tolerance⌬iRis larger than left customer’s semi-tolerance⌬iL. Therefore, the left quality

loss coefficient kiLis larger than the right quality loss coefficient Table 8 Normal distribution and lognormal distribution

veri-fied by chi-square test

Interval 共mm兲 Real number ni Theoretical number ei chi-square value 共ni− ei兲2/ ei

Normal Lognormal Normal Lognormal

⬍4.920 1 1.16 1.74 0.022 0.315 4.920–4.930 1 1.97 2.19 0.478 0.647 4.930–4.940 3 3.40 3.32 0.047 0.031 4.940–4.950 7 4.36 4.72 1.599 1.101 4.950–4.960 3 4.11 2.87 0.300 0.006 4.960–4.970 2 2.84 2.66 0.248 0.164 ⬎4.970 3 2.16 2.50 0.327 0.100 Sum 20 20.0 20.0 3.021 2.364

Table 9 Results of normal distribution with standard deviation of 0.01 mm and zero correlation coefficients

Variable Optimum standard deviation共mm兲 Lower limit of tolerance共mm兲 Upper limit of tolerance共mm兲 Total tolerance 共mm兲 x1 0.03788 0.0921 0.0921 0.1842 x2 0.01461 0.0355 0.0355 0.0710 x3 0.03168 0.0766 0.0776 0.1542 x4 0.02510 0.0610 0.0611 0.1221 x5 0.03078 0.0744 0.0753 0.1497 x6 0.02423 0.0589 0.0590 0.1179 x7 0.00753 0.0183 0.0183 0.0366 x8 0.00640 0.0155 0.0156 0.0311

Optimum machining cost Cmis 5.1128

Quality loss expected value E共Lasm兲 is 0.3888

Total cost Ctis 5.5017

The result of Monte Carlo simulation is 93.71% by 10,000 samples

Table 10 Functional dimensions with optimal tolerances specified

Variable Dimension description Dimension共mm兲

x1 Stator length 152± 0.092

x2 Rotor slot length 154.8± 0.036

x3 Rear cover plate wall thickness with cut off 5 + 0.077

−0.078

x4 Rear cover plate thickness 20± 0.061

x5 Front cover plate thickness with cut off 5 + 0.074 −0.075

x6 Front cover plate thickness 30± 0.059

x7 Vane thickness 4.95± 0.018

x8 Rotor slot width 5 ± 0.016

Fig. 15 Upper and lower tolerances of function dimensions

Fig. 16 Relation between correlation coefficient of cover plate thickness and cost

Fig. 17 Relation between standard deviations of mass produc-tion parts and cost

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KiRas shown in Table 6. The quality loss coefficients of x2and x8 are much higher than those of other dimensions because the stator and the rotor are the most expensive components in the rotary compressor.

The chi-square value 2.364 of the lognormal distribution is smaller than the value 3.021 of the normal distribution as shown in Table 8. It demonstrates that the lognormal distribution has better fitting than the normal distribution for the vane thickness when mean-shift and asymmetry exist. The mean value of vanes as shown in Table 7 is 4.947 mm and the nominal dimension as shown in Table 4 is 4.95 mm. The mean-shift value of the lognor-mal distribution due to machining is 0.003 mm, which is used for tolerance optimization in this study.

Design functions control tolerances of the vane thickness and the slot width, and the slot length affects clearances of the front cover plate and the rear cover plate. The optimization generates tighter tolerances for the slot length x2, the vane thickness x7and the slot width x8, which are critical function dimensions for manu-facturing, than other functional dimensions as shown in Fig. 15. The unit value of 0.01 mm is used in Table 10 in order to match the precision of the regular machining process.

When correlation coefficients␳x

3x4of x3and x4and␳x5x6of x5

and x6vary, the cost variation is insignificant as shown in Fig. 16. One reason is that dimension variables x3and x4, and x5and x6 have larger tolerances than others. The variation of correlation coefficients does not affect the total cost significantly. When the standard deviation of standard parts increases, the total cost in-creases as shown in Fig. 17. It means that other components of the assembly made by batch production need to have high manufac-turing accuracy in order to compensate for poor quality mass pro-duction parts.

The quality loss in the total cost function is small compared with the total cost as shown in Figs. 16 and 17. The effect of the weighting ratio Wq/ Wmvaried from 1 to 8 on the cost is

investi-gated. The results as shown in Fig. 18 indicate that the cost of quality loss is reduced when the weighting ratio is increased. The total cost is increased because tight tolerance allocation increases the machining cost. However, the situation can be changed if the part is made of high-cost material and the replacement cost of a component is much greater than the machining cost.

Conclusion

An optimization model has been established with consideration of the required functional reliability, the minimum machining cost and quality loss in this study. The procedure to obtain optimal customer’s semi-tolerance with the combined model is revealed. The reliability index model is capable of handling the normal distribution and the non-normal distribution. Tolerance allocation of the compressor is optimized. The data of the vane thickness is better fitted by the lognormal distribution than the normal distri-bution, and it is verified by chi-square test.

The oriented relationship functional graph has been used to analyze the dimension connection of the components in the slid-ing vane rotary compressor. Through the graph analysis, the tional dimension chains are sorted out and then the design func-tions for optimization are generated. Optimal tolerance allocation of functional dimensions in the rotary compressor is obtained by the non-linear programming with the efficient beta algorithm关35兴. According to the optimization results, tolerances of the rotor slot length, the rotor slot width, and the vane thickness are tighter than those of other functional dimensions. Furthermore, the effect of correlation coefficients has been analyzed. The effects of the cor-relation coefficients of the bottom cover plate and the top cover plate, and the front cover plate and the rear cover plate on the cost are insignificant. When the standard deviation of standard parts increases, the total cost increases because other components of the assembly made by batch production need to have high manufac-turing accuracy in order to compensate for poor quality mass pro-duction parts. The cost of quality loss is reduced when the

weight-ing ratio is increased. The total cost is increased because tight tolerance allocation increases the machining cost.

Acknowledgments

The authors would like to express their sincere thanks to the National Science Council of the Republic of China for Grant No. NSC 93-2218-E-002-063 for funding this investigation and to Grant D. Huang for comments and revisions made on this manu-script. Nomenclature A ⫽ replacement cost C ⫽ cost E ⫽ expected G ⫽ design function

J ⫽ Jacobian coordinates transform matrix k ⫽ quality loss coefficient

L ⫽ quality loss

m ⫽ nominal and target value n ⫽ number of components p ⫽ number of design functions V ⫽ covariance matrix

X ⫽ dimension vector x ⫽ dimension variable

y ⫽ normalized dimension variable

W ⫽ weighting coefficient of objective function Z ⫽ vector of independent standard distribution

variable

z ⫽ independent standard distribution variable ␤ ⫽ reliability index of design function ␮ ⫽ mean value of dimension

␴ ⫽ standard deviation ⌫ ⫽ set of standard deviation

␦ ⫽ coefficient of variation ⌬ ⫽ specific semi-tolerance ␳ ⫽ correlation coefficient Subscripts 0 ⫽ initial condition 1 , 2 , . . . , n ⫽ dimension asm ⫽ assembly h ⫽ hth component i ⫽ ith dimension j ⫽ jth dimension

ij ⫽ relation between ith and jth dimension k ⫽ kth design function

L ⫽ left side of mean value m ⫽ machining

min ⫽ minimum q ⫽ quality loss

R ⫽ right side of mean value t ⫽ total

x ⫽ dimension variables

z ⫽ normalized dimension variable Superscripts

* ⫽ required ¯ ⫽ mean

t ⫽ transport Appendix

The beta algorithm that was developed by Hohenbichler and Rackwitz关35兴 is an iteration method to calculate the reliability index. The steps for the normal distribution or the lognormal dis-tribution are listed as follows关40兴:

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b. Using Rosenblatt transformation to transfer X0 into an independent normalized vector Z0.

c. Determine the Jacobian matrix

J =⳵共z1,z2,… . ,zn⳵共x1,x2,… . ,xn兲 =

⳵z1 ⳵x1 ⳵z1 ⳵x2 ¯ ⳵z1 ⳵xn ⳵z2 ⳵x1 ⳵z2 ⳵x2 ¯ ⳵z2 ⳵xn ] ]  ] ⳵zn ⳵x1 ⳵zn ⳵x2 ¯ ⳵zn ⳵xn

共A1兲

d. Obtain the value of the design function G and its gradient ⵜG

Gz共Z0兲 = GX共X0兲 共A2兲

ⵜGZ=共J−1兲tⵜ GX 共A3兲

e. The new point can be obtained from the gradient of the vector Z*= 1 ⵜGZ t ⵜ GZ 关ⵜGZ t Z0− GZ共Z0兲兴 ⵜ GZ 共A4兲

f. The new iteration point in X space can be approached by the first-order linear approximation

X*⬵ X0+ J−1共Z*− Z0兲 共A5兲 g. Obtain the reliability index for this iteration

␤ = 共Z*tZ*1/2 共A6兲

h. Input X*into the iteration process again until the

conver-gence is reached. References

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數據

Fig. 1 Components and schematic drawing of sliding vane ro- ro-tary compressor
Fig. 4 Asymmetric quality loss curve and asymmetric dimen- dimen-sion distribution
Fig. 8 Oriented functional relationship graph of rotary compressor
Fig. 10 Second functional dimension chain
+3

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