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Tolerance charting for components with both angularand square shoulder featuresJianbin Xue a & Ping Ji aa Department of Industrial and Systems Engineering , The Hong Kong Polytechnic University ,Hung Hom, Kowloon, Hong Kong E-mail:Published online: 23 Feb 2007.

To cite this article: Jianbin Xue & Ping Ji (2005) Tolerance charting for components with both angular and square shoulderfeatures, IIE Transactions, 37:9, 815-825, DOI: 10.1080/07408170590969843

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IIE Transactions (2005) 37, 815–825Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/07408170590969843

Tolerance charting for components with both angularand square shoulder features

JIANBIN XUE and PING JI∗

Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong KongE-mail: mfpji@polyu.edu.hk

Received December 2004 and accepted April 2005

This paper deals with tolerance charting for a component with both angular and square shoulder features. First of all, a unifiedsurface model is proposed for a component with both angular and square-shoulder-featured surfaces. Then, a two-dimensionaltolerance chart is constructed and parameterized for the machining of the component. Also, two general formulae are derived tocalculate the surface changes due to machining cuts. The recording of the surface changes during all machining cuts means that thereverse dimensional chains and the “solid” stock removals can be identified automatically. Finally, the forward dimensional chainsare obtained by inverting the reverse dimensional chains. Both the reverse and forward dimensional chains developed in this paperare two-dimensional, with both axial and radial dimensions. The approach presented in this paper is very useful in the developmentof a computer-aided angular tolerance charting system.

1. Introduction

Computer-Aided Process Planning (CAPP) is of great im-portance in computer-integrated manufacturing systems.It is a bridge between Computer-Aided Design (CAD) andComputer-Aided Manufacturing (CAM). In terms of thedimensions of a product (or component), the output dataof a CAD system are blueprint dimensions whereas the in-put data for a CAM system are working dimensions. TheCAD blueprint dimensions are designed based on functionsor assembly specifications, whereas the CAM working di-mensions are considered from a manufacturing perspective.CAPP consists in designing a sequence of operations tomachine a desired component considering both function-ality and manufacturability. Tolerance charting, as used inthe process planning, is usually applied to derive the CAMworking dimensions for the machining cuts from the CADblueprint dimensions. The essential ingredients of the toler-ance charts are the dimensional chains (also called tolerancechains) among the machining cuts. Considerable attentionhas been focused on the study of tolerance charting, how-ever, most of the reported literature concerns square shoul-der features only, that is, one-dimensional (1-D) tolerancecharting. When angular features are involved, tolerancecharting becomes much more complex. This paper presentsa new approach to deal with angular tolerance charting.Consequently, as angular features are in a two-dimensional

∗Corresponding author

(2-D) space domain, the 1-D tolerance charting for squareshoulder features is a special case of the angular tolerancecharting.

2. Literature review

Tolerance charts have been in use in manufacturing indus-tries for about 50 years. Early tolerance charts (Mooney,1955; Eary and Johnson, 1962; Wade, 1967; 1983) weremanually constructed and calculated. The manual chart-ing process is tedious, time-consuming, and error-prone.Since the 1980s, several computer-aided tolerance chartingsystems (Sack, 1982; Ahluwalia and Karolin, 1984; Tangand Davies, 1988; Irani et al., 1989) have been developed.These are 1-D tolerance charting systems that deal withcomponents (workpieces) with square shoulder features. A2-D angular tolerance charting system that is able to dealwith workpieces with angular features and other features,is a relatively new development especially for computer-aided angular tolerance charting. So far, only a few studieshave been performed on computer-aided angular tolerancecharting. Two papers on manual angular tolerance chartinghave been presented by Wade (1967, 1983). The angular ma-chining cuts are grouped into different classes using processsequences, blueprint setups, and the movement of a controlpoint relative to the datum. Different trigonometric formu-lae for 42 “standard” cases were derived to calculate theworking dimensions and their tolerances for angular ma-chining cuts. Based on Wade’s work, a rule-based systemfor angular tolerance charting was developed by Nee and

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Senthil Kumar (1992), in which various IF-THEN ruleswere set up in order to select appropriate formulae to com-pute working dimensions and their tolerances for angularmachining cuts. Pan and Tang (2001) highlighted the useof dummy cuts to reflect the corresponding dimensionalchanges due to the indirect effect when the angular sur-faces are machined. Although the 28 most frequently usedformulae were reorganized into eight groups, Wade’s idea ofselecting appropriate trigonometric formulae was still fol-lowed. In summary, these previous studies were all based onWade’s 42 trigonometric formulae, and it is quite difficultfor a computer to select the correct formulae for a specificcase among so many cases. If the case is not standard, thesituation becomes more complicated since even a manualsolution may not exist.

Recently, a 2-D tolerance chart was constructed for angu-lar features by Xue and Ji (2001). Also, an algebraic method,initially proposed to identify dimensional chains in 1-D tol-erance charts by Ji (1999) has been extended to the angulartolerance charting of three different cases by Ji and Xue(2002). The process tolerance allocation problem in 2-Dtolerance charting was discussed by Xue and Ji (2004). Thispaper presents a unified surface model to describe both theangular and square-shoulder-featured surfaces regardlessof Wade’s classification of angular cuts, and two generaltrigonometric formulae are derived to identify the surfacechanges caused by the machining cuts. The set of two for-mulae are universal, cover all of Wade’s 42 trigonometricformulae as well as other nonstandard situations for an-gular features, and work not only for angular features butalso square shoulder features. The 2-D chains include bothaxial and radial dimensions and are obtained from workingdimension equations which are set up from the state of thesurfaces at each stage of machining in the 2-D parameter-ized tolerance chart. The approach discussed in this papercan also identify the “solid” stock removals, and removethem. This approach is a useful approach to the develop-ment of a computer-aided tolerance charting system forboth angular and square-shoulder-featured components.

3. Preliminaries

A workpiece (component) with a tapered cylindrical sur-face shown in Fig. 1 is taken as an example to describethe surfaces in the unified surface model and to derive thetwo trigonometric formulae to identify the surface changescaused by machining cuts. The 2-D angular tolerance chartin Fig. 2 for the workpiece is used to illustrate how to findout the dimensional chains and determine the “solid” stockremovals.

3.1. Describing the surface model in a suitable coordinatesystem

In order to clearly describe the surfaces of the workpiece,a coordinate system should be set up. Since the angular

Fig. 1. A workpiece with angular features.

features result in a 2-D space, a 2-D coordinate system is setup as shown in Fig. 1. The centerline of the workpice is setup as the x-axis with the leftmost surface of the workpiecebeing the y-axis, and their intersection is the origin of thecoordinate system. The rightward axial direction is takento be the positive direction of the x-axis and the downwardradial direction is taken to be the positive direction of they-axis. Since the workpiece is symmetrical considering halfof the workpiece is sufficient to represent a tolerance chart.

The features discussed in this paper are either angularor square shoulder features. A square shoulder surface isdefined as a surface that is parallel either to the x-axis or tothe y-axis whereas an angular surface is defined as a surfacethat is neither parallel to the x-axis nor the y-axis. Angu-lar features include the angular surface and its adjacentsurfaces whereas square shoulder features are those squareshoulder surfaces not adjacent to any angular surface.

The workpiece in Fig. 1 has six surfaces. Surface D is anangular surface. Surfaces B, C, E, and F are square shouldersurfaces. However, only surfaces B and E are square shoul-der features, whereas surfaces C and F are angular featuressince they are adjacent to angular surface D. Surface A isa pseudo-surface that functions as the datum of surfaces Band C.

Each surface can be fully described with a line segmentand its normal in 2-D space, as shown in Fig. 3. A is thenormal vector of the surface being considered. The twoend-points of the line segment represent the delimitations ofthe surface. The surface’s angle, between the positive x-axisdirection and the surface’s normal, represents the directionof the surface. The angle is measured from the x-axis to they-axis with the right-hand rule. Now, the surface model forthe surface in Fig. 3 is represented by ((X1, Y1), (X2, Y2),α).

By applying the surface model to the workpiece in Fig. 1,we can represent

surface B as B ((X1, Y2), (X3, Y2), −90◦);surface C as C ((X1, Y4), (X2, Y4), 90◦);surface D as D ((X2, Y4), (X3, Y3), 90◦–θ );surface E as E ((X1, Y1), (X1, Y4), 180◦);surface F as F ((X3, Y1), (X3, Y3), 0◦);

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Fig. 3. A surface model in 2-D space.

Surface A is a special surface. It can be expressed as A((X1, Y1), (X3, Y1), 90◦). Its angle can be either 90◦ or270◦, and 90◦ is preferred since an angle can be defined tobe within ±180◦.

The angles are assumed to be constant to simplify the an-gular tolerance charting (Wade, 1967, 1983). A change ina surface can be represented by changes in the coordinatesof the two end-points of the surface in the 2-D coordinatesystem. By recording the changes in the coordinates, the sur-face changes can be equivalently identified. For the work-piece shown in Fig. 1, by recording the coordinates, that is,X1, X2, X3 and Y1, Y2, Y3, Y4, the surface changes ateach machining cut can be closely determined.

3.2. Surface signs

Manufacturing is a process in which the shapes and di-mensions of the raw material are changed into the finalproduct in a step-by-step manner. The surfaces are cut tocreate the desired shapes and dimensions, and thus the sur-faces change during machining. Surface changes can be ex-pressed in terms of the corresponding coordinate changesin the surface model, as mentioned earlier. For example,in Fig. 1, if a machining cut is made on surface F, X3 willchange. Before the machining cut, X3 should have a stockremoval, that is, X3 should increase in the positive directionof the x-axis. In this case, X3 is positive. Ji (1999) definedthe rules for determining surface signs for square shouldersurfaces. When a machining cut is made on a surface, if thesurface change increases along the positive direction of thex-axis, the surface’s sign is positive (+), and if the surfacechange decreases along the positive direction of the x-axis,it is negative (–). The rules are applicable to both externaland internal surfaces in 1-D tolerance charting. Althoughthe rules only consider the surfaces along the x-axis (axialsurfaces), obviously, they can also be applied to the surfacesalong the y-axis (radial surfaces). The use of these rules canlead to the directions of the square shoulder surfaces in Fig.1 being determined. Surface E is negative, represented by−X1 whereas surface F is positive, represented by +X3.Surface B is negative, represented by −Y2 whereas surfaceC is positive, represented by +Y4.

When angular surfaces are taken into account, these rulesare no longer applicable. When a cut is made on surface D,both X2 and Y4 will change, now it is difficult to determineif the surface is increasing (+) or decreasing (−). As forthe angular features, the surface signs must be considereddifferently. For example, when a machining cut is made onsurface C, Y4 will change, and it is positive (+Y4) accordingto the above rules. At the same time, X2 also changes and itis negative (–X2). However, when a machining cut is madeon surface D, X2 will increase in the positive direction of thex-axis. According to the rules, X2 is positive (+X2) in thiscase. The dilemma here is that X2 is positive or negative, andY3 has the same dilemma. The signs of X2 and Y3 cannotbe determined by the above rules. It is the angular featurethat causes this dilemma. When a machining cut is madeon an angular surface or its adjacent surfaces, that is, theangular features, two or more coordinates will change, andtheir coordinates’ signs cannot be easily determined. Thisis the big difference between machining cuts on the squareshoulder features and the angular features. To tackle thedilemma, a new approach must be developed.

3.3. Trigonometric formulae for changes inmachined surfaces

Using the surface model described earlier it is possible to de-termine the surface changes regardless of the surface signs.The normals of the angular surface and its adjacent surfacesdetermine the direction (or sign) of the surface changes.The surface changes can be represented as the coordinatechanges of the surface’s two end-points. If one of the twoend-points has its coordinate changes as �x and �y, re-spectively, the following two trigonometric formulae arecapable of determining the changes of the machined sur-faces for both angular and square shoulder features, andfor both internal and external surfaces:

�x = −a sin(β)sin(α − β)

, �y = a cos(β)sin(α − β)

, (1)

where a is the stock removal in the direction of the ma-chined surface’s normal; α is the angle between the positivex-axis and the machined surface’s normal; and β is the an-gle between the positive x-axis and the adjacent surface’snormal.

The α and β angles must be calculated from the positivex-axis to the normal of the surface (whatever the angularsurface or its adjacent surface) with the right-hand rule.The derivation of the set of two formulae is illustrated inAppendix A. Tables A1 to A3 in Appendix B show severaltypical cases considered using the two formulae, includingthe cases where the surface is internal, external, angular andsquare shoulder featured. The set of two formulae has beenverified to be applicable to Wade’s 42 standard cases (Xue,2003), even for a “V” shaped angular feature. As a result,the set of two formulae is universal.

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Tolerance charting for multi-feature components 819

Table 1. Surface changes in the tolerance chart

Surface changes

SQ MS

Workingdimensionequation X1 X2 X3 Y1 Y2 Y3 Y4

B/P 0 B1 − B2 B1 0 B3 B4 − B2 × tan(θ ) B410 C W10 = Y4 − Y1 = B4 B1 − B2 − S10 ×

cot(θ )B4 + S10

9 D W9 = X2 − X1 = B1 −B2 − S10 × cot(θ )

B1 − B2 − S10 ×cot(θ ) + S9 ×csc(θ )

B4 − B2 × tan(θ ) +S9 × sec(θ )

8 F W8 = X3 − X1 = B1 B1 + S8 B4 − B2 × tan(θ ) +S9 × sec(θ ) −S8 × tan(θ )

7 B W7 = Y2 − Y1 = B3 B3 − S76 D W6 = X2 − X1 =

B1 − B2 − S10 ×cot(θ ) + S9 × csc(θ )

B1 − B2 − S10 ×cot(θ ) + S9 ×csc(θ ) + S6 ×csc(θ )

B4 − B2 × tan(θ ) +S9 × sec(θ ) −S8 × tan(θ ) +S6 × sec(θ )

5 C W5 = Y4 − Y1 =B4 + S10

B1 − B2 − S10 ×cot(θ ) + S9 ×csc(θ ) + S6 ×csc(θ ) − S5 ×cot(θ )

B4 + S10 +S5

4 F W4 = X3 − X1 =B1 + S8

B1 + S8 + S4 B4 − B2 × tan(θ ) +S9 × sec(θ ) −S8 × tan(θ ) +S6 × sec(θ ) −S4 × tan(θ )

3 B W3 = Y2 − Y1 =B3 − S7

B3 − S7 − S3

2 C W2 = Y4 − Y1 =B4 + S10 + S5

B1 − B2 − S10 ×cot(θ ) + S9 ×csc(θ ) + S6 ×csc(θ ) − S5 ×cot(θ ) − S2 ×cot(θ )

B4 + S10 +S5 + S2

1 E W1 = X3 − X1 =B1 + S8 + S4

−S1

4. Identification of the dimensional chainsin the 2-D tolerance chart

Before applying the surface model and the two universalformulae to a tolerance chart, the tolerance chart of theworkpiece should be parameterized, as shown in Fig. 2. Inthe parametric tolerance chart, all surfaces of the work-piece should be represented with their corresponding co-ordinates, Xi and Yi. Xi represents an axial coordinatewhereas Yi represents a radial coordinate. All Yi are re-flected by a mirror so as to be parallel to Xi in the toler-ance chart, as shown in Fig. 2. The working dimensionsare parameterized as Wi, the stock removals as Si, theblueprint dimensions as Bi, and the resultant dimensionsas Ri. The tolerances of these dimensions can be repre-sented with their corresponding lowercase letters, that is,

wi, si, bi, and ri, respectively. With such a parametric toler-ance chart, the working dimensions and their tolerancescan be manipulated freely without consideration of realdata.

In tolerance charting, one of the main tasks is to identifythe dimensional chains. Once the dimensional chains arefound, the other two main tasks, namely the determinationof working dimensions and allocation of process tolerances,will be much easier. This section discusses how to use thesurface model and the two universal formulae to identifythe dimensional chains in the angular tolerance charting.

First of all, the initial working dimensions should be de-termined from the blueprint dimensions. In order to identifythe surface changes during the machining cuts, the initialcoordinates of each surface must be obtained first. Fromthe blueprint dimensions Bi (i = 1, 2, 3, 4) in Fig. 2, the

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following equation set can be found:

B1 = X3 − X1,

B2 = X3 − X2,

B3 = Y2 − Y1,

B4 = Y4 − Y1,

tan(θ ) = Y4 − Y3X3 − X2

,

Y1 = 0,

X1 = 0.

(2)

The first four equations are directly derived from theblueprint dimensions. The fifth equation is the expressionfor the angle θ . The last two equations are from the origin ofthe coordinate system. Now, Xi and Yi can be determinedeasily from the linear equation set (2) as follows:

X1 = 0,

X2 = B1 − B2,

X3 = B1,

Y1 = 0,

Y2 = B3,

Y3 = B4 − B2 × tan(θ ),Y4 = B4.

(3)

After the initial values of Xi and Yi (in fact, they are the fi-nal values in the product) are calculated, the coordinates ofeach surface during machining can be calculated backwardsfrom the last machining cut to the first one. As long as thesurface changes are clearly identified, the working dimen-sion equations (to be discussed later) can be constructedeasily and correctly.

Table 1 shows the surface changes of the workpiece. “SQ”is the abbreviation for Sequence Operation and “MS” isfor Machined Surface. The values in row “B/P” (blueprint)are derived from the blueprint dimensions, that is, Equation(3), the initial values. They are the final surface status valuesafter all machining cuts are made.

In operation cut 10, the working dimension is W10, andshould be measured between surfaces Y1 and Y4 after thecut, consequently, its value is Y4 − Y1. In other words,W10 = Y4 − Y1 = B4 − 0 = B4. This represents a workingdimension equation.

Also, in this operation, a machining cut is made on sur-face C with the stock removal of S10. According to thesurface model, X1 and Y4 (one end-point of the surface)will change. X2 and Y4 (the other end-point of the surface)will also change. After this operation cut, X1, X2, and Y4have to be updated.

Since surfaces E and C form X1 and Y4 and the machin-ing cut is made on surface C, the angle of surface C is 90◦in the surface model, and the angle of adjacent surface E

is 180◦, and the stock removal is S10. Now, “α” is 90◦, “β”is 180◦ and “a” is S10 in Equation (1), and the changedvalues of X1 and Y4 are obtained as follows:

�X1 = −a sin(β)sin(α − β)

= −S10 × sin(180◦)sin(90◦ − 180◦)

= 0,

�Y4 = a cos(β)sin(α − β)

= S10 × cos(180◦)sin(90◦ − 180◦)

= S10.

Similarly, surfaces C and D are adjacent, the machiningcut is made on surface C, the angle of surface C is 90◦ (=α),and the angle of surface D is 90◦ − θ (=β), and the stockremoval is still equal to S10 (=a). The changed value of X2and Y4 are as follows:

�X2 = −a sin(β)sin(α − β)

= −S10 × sin(90◦ − θ )sin(90◦ − (90 − θ ))

= −S10 × cot(θ ),

�Y4 = a cos(β)sin(α − β)

= S10 × cos(90◦ − θ )sin(90◦ − (90◦ − θ ))

= S10.

It is very interesting to notice that although Y4 has beenapplied by Equation (1) twice, the two values are the same,and certainly, they must be the same.

The changes in X1, X2 and Y4 in operation 10 have alsobeen obtained. The status values of each surface should beupdated. The values in row 10 are the surface status beforemachining cut 10 is made, that is, after machining cut 9is made. These values are recorded in the row of SQ 10 inTable 1.

The working dimension equations and the surfacechanges in other machining cuts, from SQ 9 to SQ 1, canalso be obtained one by one with the same procedure, asshown in Table 1. In summary, the following equation setfor the working dimensions can be obtained:

W1 = B1 + S8 + S4.

W2 = B4 + S10 + S5.

W3 = B3 − S7.

W4 = B1 + S8.

W5 = B4 + S10.

W6 = B1 − B2 − S10 × cot(θ ) + S9 × csc(θ ).W7 = B3.

W8 = B1.

W9 = B1 − B2 − S10 × cot(θ ).W10 = B4.

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Tolerance charting for multi-feature components 821

The above equations can be rewritten in matrix formatwith all working dimensions, blueprint dimensions andstock removals as follows:

W1W2W3W4W5W6W7W8W9W10

=

0 0 0 1 0 0 0 1 0 0 1 0 0 00 0 0 0 1 0 0 0 0 1 0 0 0 10 0 0 0 0 0 −1 0 0 0 0 0 1 00 0 0 0 0 0 0 1 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 10 0 0 0 0 0 0 0 csc(θ ) − cot(θ ) 1 −1 0 00 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 − cot(θ ) 1 −1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 1

×

S1

S2

S3S4S5

S6S7S8S9

S10B1B2B3B4

. (4)

In the big matrix of Equation (4), the elements of columns1, 2, 3, and 6 are all zeroes. Their counterparts, S1, S2,S3, and S6, have no effect on the working dimensions.These four stock removals are called “solid.” By eliminatingthese four “solid” stock removals and their correspondingcolumns in the matrix, Equation (4) becomes:

W1W2W3W4W5W6W7W8W9W10

=

1 0 0 1 0 0 1 0 0 00 1 0 0 0 1 0 0 0 10 0 −1 0 0 0 0 0 1 00 0 0 1 0 0 1 0 0 00 0 0 0 0 1 0 0 0 10 0 0 0 csc(θ ) − cot(θ ) 1 −1 0 00 0 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 0 00 0 0 0 0 − cot(θ ) 1 −1 0 00 0 0 0 0 0 0 0 0 1

×

S4S5S7S8S9

S10B1B2B3B4

. (5)

S4S5S7S8S9

S10B1B2B3B4

=

1 0 0 −1 0 0 0 0 0 00 1 0 0 −1 0 0 0 0 00 0 −1 0 0 0 1 0 0 00 0 0 1 0 0 0 −1 0 00 0 0 0 0 sin(θ ) 0 0 − sin(θ ) 00 0 0 0 1 0 0 0 0 −10 0 0 0 0 0 0 1 0 00 0 0 0 − cot(θ ) 0 0 1 −1 cot(θ )0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1

×

W1W2W3W4W5W6W7W8W9

W10

. (6)

This equation set presents the reverse dimensional chains.The concepts of reverse dimensional chains and forward

dimensional chains were discussed earlier (Ji et al., 1996).By inverting the reverse matrix, the forward dimensionalchains can be obtained as follows in Equation (6):

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Once the reverse and forward dimensional chains arefound, the tolerance chart becomes a very powerful tool towork out the unknown working dimensions and their tol-erances required in process planing. If the stock removals(Si) and blueprint dimensions (Bi) are known, the work-ing dimensions (Wi) can be calculated with the reverse di-mensional chains. This is obvious from Equation (5). Theforward dimensional chains can be used to allocate toler-ances to machining cuts if they are not known or to checkwhether the tolerances are suitable if they are assigned man-ually. This has been discussed by many researchers (Iraniet al., 1989; Ji et al., 1996). However, it should be noticedthat the tolerance allocation is more difficult for a workpiecewith an angular feature due to more complicated forwarddimensional chains.

In summary, the approach is very simple. First of all,the initial coordinates of the surfaces are obtained fromthe blueprint dimensions. Then in the main process of theapproach, a working dimension equation is constructed foreach machining cut. For the same machining cut, the twouniversal formulae should be applied twice to the two end-points of the machined surface so that the surface changesare identified. This operation is performed on all machiningcuts, one by one, and from the last cut to the first one. Oncethis process is completed, all working dimension equationsare identified, and they are the reverse dimensional chains.The forward dimensional chains are obtained by invertingthe reverse matrix.

5. Remarks

In this paper, the dimensional chains are obtained from aparametric tolerance chart with the surface model. Eachrow of the matrix represents a dimensional chain. Can thesedimensional chains be identified using a traditional methodfor 1-D tolerance charting, such as the manual tracingmethod (Wade, 1983) or the computerized tracing method(He and Lin, 1992)? Among the dimensional chains in theexample, B4, B3, and B1 can be expressed in 1-D mode andthe traditional tracing method may be used. However, B2is a complex case, which has to be considered in 2-D mode.In other words, B2 is a 2-D dimensional chain. BlueprintB2 = − cot(θ )W5 + W8 − W9 + cot(θ )W10 in Equation(6). This forward dimensional chain includes not only thex-axial dimensions, W8 and W9, but also the y-radial di-mensions, W5 and W10. This concept is also applicable toa reverse dimensional chain. For example, W6 in Equation(5) is a 2-D dimensional chain. Attempts have been madeto identify a 2-D dimensional chain in angular tolerancecharting by use of different traditional methods, such asthe tracing method, but few of them have been successfullyused.

6. Conclusions

This paper has presented a surface model and two univer-sal formulae for the tolerance charting of both angular andsquare shoulder features. An approach to identify the 2-Ddimensional chains by use of the model and the formu-lae was highlighted. The presented approach works for allof Wade’s 42 standard cases and also nonstandard cases,and for both angular and square shoulder features, so itis universal and effective. Also, we highlight the point thatthe approach is very simple to apply. The developed modeland approach are a useful step towards the developmentof a computer-aided angular tolerance charting system forboth angular and square shoulder featured components inCAPP.

Acknowledgement

This work described in this paper was supported by theResearch Grants Council of Hong Kong, China (projectPolyU 5142/98E).

References

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Appendices

Appendix A

Without loss of generality, the surfaces involved are as-sumed to be angular featured surfaces. A general modelof angular cuts is shown in Fig. A1.

Fig. A1. A general model of angular machining cuts.

Surfaces PF and PA are the angular surfaces being con-sidered. M is the normal of surface PA, and N is the normalof surface PF . The angle between the positive x-axis andnormal M is β, and the angle between the positive x-axisand normal N is α. Both surfaces have the coordinates ofpoint P in common. When a machining cut with an amountof stock removal “a” is made on surface PF , point P willmove to point Q, which indicates the changes of the surface.Here, the formulae to calculate the coordinates’ changes arederived as follows:

· ·· � QPD = 180◦ − � PAB = 180◦ − (90◦ + β) = 90◦ − β,

· ·· � FPD = 180◦ − � HFP = 180◦ − (α − 90o) = 270◦ − α,

·· ·� EPQ = � FPD − 90◦ − � QPD

= 270◦ − α − 90◦ − (90◦ − β) = 90◦ − α + β,

· ·· PE = a,

· ·· PE/PQ = cos(� EPQ) = cos(90◦ − α + β) = sin(α − β),

·· ·PQ = PE/ sin(α − β) = a/ sin(α − β),

· ·· �x = PQ × cos(� QPD),

·· ·�x = asin(α − β)

× cos(90◦ − β) = a sin(β)sin(α − β)

,

· ·· �y = PQ × sin( � QPD),

·· ·�y = asin(α − β)

× sin(90◦ − β) = a cos(β)sin(α − β)

.

By deriving backwards, before the machining cut, x willdecrease with �x, and y will increase with �y. So, the lasttwo formulae should be written as follows:

�x = −a sin(β)sin(α − β)

,

�y = a cos(β)sin(α − β)

.

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824 Xue and Ji

Appendix B

Table A1. Typical machining cuts on internal angular featured surfaces

Feature type Given Direct results Results from the formulae

Cut on W with stock “a” U: 180◦ (β)W: −90◦ − θ (α)

�xp = 0�yp = −a/cos(θ )

�xp = −a sin(β)sin(α−β) = 0

�yp = a cos(β)sin(α−β) = −a

cos(θ )

V: 0◦ (β)W: −90◦ − θ (α)

�xq = 0�yq = −a/cos(θ )

�xq = −a sin(β)sin(α−β) = 0

�yq = a cos(β)sin(α−β) = −a

cos(θ )

Cut on U with stock “a” U: 180◦ (α)W: −90◦ − θ (β)

�xp = −a�yp = a× tan(θ )

�xp = −a sin(β)sin(α−β) = −a

�yp = a cos(β)sin(α−β) = a tan(θ )

U: 180◦ (α)V: 90◦ (β)

�xq = −a�yq = 0

�xq = −a sin(β)sin(α−β) = −a

�yq = a cos(β)sin(α−β) = 0

Cut on U with stock “a” U: 0◦ (α)W: −90◦ − θ (β)

�xp = +a�yp = −a× tan(θ )

�xp = −a sin(β)sin(α−β) = a

�yp = a cos(β)sin(α−β) = −a tan(θ )

U: 0◦ (α)V: 90◦ (β)

�xq = +a�yq = 0

�xq = −a sin(β)sin(α−β) = a

�yq = a cos(β)sin(α−β) = 0

Table A2. Typical machining cuts on external angular featured surfaces

Feature type Given Direct results Results from the formulae

Cut on W with stock “a” W: 90◦ − θ (α)U: 180◦ (β)

�xp = 0�yp = a/cos(θ )

�xp = −a sin(β)sin(α−β) = 0

�yp = a cos(β)sin(α−β) = a

cos(θ )

W: 90◦ − θ (α)V: 0◦ (β)

�xq = 0�yq = a/cos(θ )

�xq = −a sin(β)sin(α−β) = 0

�yq = a cos(β)sin(α−β) = a

cos(θ )

Cut on V with stock “a” V: 0◦ (α)W: 90◦ − θ (β)

�xp = +a�yp = a× tan(θ )

�xp = −a sin(β)sin(α−β) = a

�yp = a cos(β)sin(α−β) = a tan(θ )

V: 0◦ (α)A: 90◦ (β)

�xq = +a�yq = 0

�xq = −a sin(β)sin(α−β) = a

�yq = a cos(β)sin(α−β) = 0

Cut on U with stock “a” U: 180◦ (α)W: 90◦ − θ (β)

�xp = −a�yp = a× tan(θ )

�xp = −a sin(β)sin(α−β) = −a

�yp = a cos(β)sin(α−β) = a tan(θ )

U: 180◦ (α)A: 90◦ (β)

�xq = −a�yq = 0

�xq = −a sin(β)sin(α−β) = −a

�yq = a cos(β)sin(α−β) = 0

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Tolerance charting for multi-feature components 825

Table A3. Typical machining cuts on square-shoulder-featured surfaces

Feature type Given Direct results Results from the formulae

Cut on W with “a” W: 180◦ (α)V: −90◦ (βv)U: 90◦ (βu)

�xp = −a�yp = 0�xq = −a�yq = 0

�xp = −a sin(βv)sin(α−βv) = −a

�yp = a cos(βv)sin(α−βv) = 0

�xq = −a sin(βu)sin(α−βu) = −a

�yq = a cos(βu)sin(α−βu) = 0

Cut on W with “a” W: −90◦ (α)V: 180◦ (βv)U: 0◦ (βu)

�xp = 0�yp = −a�xq = 0�yq = −a

�xp = −a sin(βv)sin(α−βv) = 0

�yp = a cos(βv)sin(α−βv) = −a

�xq = −a sin(βu)sin(α−βu) = 0

�yq = a cos(βu)sin(α−βu) = −a

Cut on U with stock “a” U: 0◦ (α)W: −90◦ (βw)V: 90◦ (βv)

�xp = +a�yp = 0�xq = +a�yq = 0

�xp = −a sin(βw)sin(α−βw) = a

�yp = a cos(βw)sin(α−βw) = 0

�xq = −a sin(βv)sin(α−βv) = a

�yq = a cos(βv)sin(α−βv) = 0

Cut on W with stock “a” W: 90◦ (α)V: 180◦ (βv)U: 0◦ (βu)

�xp = 0�yp = a�xq = 0�yq = a

�xp = −a sin(βv)sin(α−βv) = 0

�yp = a cos(βv)sin(α−βv) = a

�xq = −a sin(βu)sin(α−βu) = 0

�yq = a cos(βu)sin(α−βu) = a

Biographies

Jianbin Xue received his Ph.D. from the Department of Industrial andSystems Engineering, The Hong Kong Polytechnic University in 2003.Currently he is an Associate Professor in the Department of MechanicalEngineering, Nanjing University of Aeronautics and Astronautics. Hisresearch interests are in tolerancing and CAD/CAM.

Ping Ji received his Ph.D. from the Department of Industrial Engineering,West Virginia University, USA. Currently he is Associate Professor inthe Department of Industrial and Systems Engineering, The Hong KongPolytechnic University. His current research interests are in productionmanagement and CAD/CAM.

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