This article was downloaded by: [Georgia Tech Library] On: 26 June 2014, At: 14:41 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Production Research Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tprs20
Unified functional tolerancing approach for precision cylindrical components a
b
b
X. D. Zhang , C. Zhang , B. Wang & S. C. Feng a
c
Mahr Federal, Inc. , 1144 Eddy St Providence, RI 02940, USA
b
Industrial Engineering , FAMU-FSU College of Engineering , 2525 Pottsdamer Street, Tallahassee, FL 32310, USA c
Manufacturing Systems Integration Division , NIST , Gaithersburg, MD 20899, USA d
Industrial Engineering , FAMU-FSU College of Engineering , 2525 Pottsdamer Street, Tallahassee, FL 32310, USA E-mail: Published online: 22 Feb 2007.
To cite this article: X. D. Zhang , C. Zhang , B. Wang & S. C. Feng (2005) Unified functional tolerancing approach for precision cylindrical components, International Journal of Production Research, 43:1, 25-47, DOI: 10.1080/00207540412331282060 To link to this article: http://dx.doi.org/10.1080/00207540412331282060
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions
International Journal of Production Research, Vol. 43, No. 1, 1 January 2005, 25–47
Unified functional tolerancing approach for precision cylindrical components X. D. ZHANGy, C. ZHANGz*, B. WANGz and S. C. FENG}
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
yMahr Federal, Inc., 1144 Eddy St Providence, RI 02940, USA zIndustrial Engineering, FAMU-FSU College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310, USA §Manufacturing Systems Integration Division, NIST, Gaithersburg, MD 20899, USA
(Revision received March 2004) Current geometric dimensioning and tolerancing (GD&T) standards dictate that the geometry of a cylindrical manufactured part should be characterized in terms of its roundness, straightness, cylindricity and diameter. However, standards define the form errors using maximum peak-to-valley values – a very simplistic geometric description. As a result, measurements based on GD&T definitions for manufactured parts are ineffective for identifying and diagnosing error sources in a manufacturing process. This paper introduces a new tolerancing scheme for cylindrical parts and its application to functional tolerancing for diesel engine components. The new tolerance definition method is based on Legendre and Fourier polynomials for modelling and characterizing typical geometric errors found in machined cylindrical parts. As the tolerance parameters in the new tolerancing scheme represents the whole profile characteristics of the part geometry, they provide a direct link to part function and error sources of manufacturing processes. This paper describes the Legendre and Fourier polynomials-based tolerancing method. Typical machining errors of cylindrical parts in precision turning and grinding processes are analysed and modelled using the new tolerancing method. Application of the new tolerancing method is illustrated with an example of functional tolerancing a production diesel engine component, which shows that the new tolerancing method is effective for the specification and control of fuel leakage variation in diesel engine component design and manufacturing. Keywords: Geometric dimensioning and tolerancing; Functional tolerancing; Legendre and Fourier polynomials; Precision machining
1. Introduction Manufacturing process imperfections, such as machine tool spindle errors, workpiece thermal expansion, part fixturing distortion and tooling wear, can cause machined workpieces to deviate from their ideal (i.e. designed and intended) geometry. Sophisticated manufacturing processes or strict process controls are necessary to ensure that excessive deviations, which can cause part malfunction and improper product performance, do not occur. To address the problems of excessive deviation, * Corresponding author. E-mail:
[email protected] International Journal of Production Research ISSN 0020–7543 print/ISSN 1366–588X online # 2005 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00207540412331282060
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
26
X. D. Zhang et al.
inspection devices are used to measure and compare the deviation of a manufactured part against its designed tolerances. These measurement results identify the causes of the deviation and in turn control and improve the manufacturing process. Thus, a key issue in precision metrology is the question of which manufacturing process geometries should be characterized and measured. Current geometric dimensioning and tolerancing (GD&T) standards (ISO 1983, ASME 1994) dictate that the geometry of a cylindrical manufactured part should be characterized in terms of its roundness, straightness, cylindricity and diameter. However, these GD&T standards define roundness, straightness and cylindricity using maximum peak-to-valley values – a very simplistic geometric description. The values provided by these standards are even more ambiguous with respect to the diameter definition of a manufactured part. As a result, measurements based on GD&T definitions for manufactured parts are ineffective for identifying and diagnosing error sources of a manufacturing process. This paper introduces a new method based on Legendre and Fourier (L–F) polynomials for modelling and characterizing typical geometric errors found in machined cylindrical parts. This method is used to model manufactured errors in precision turning and grinding processes, which are among the most common processes used in manufacturing industry. In this research, manufactured errors and their sources were studied in three independent categories: cross-section form errors, axial form errors and cross-section size errors. For each category, the modelling of typical manufactured errors is discussed. This research develops a novel method for modelling and characterizing manufactured errors, and demonstrates that this new method can be applied to diagnose and control these manufacturing errors. This research studies the manufacturing errors that occur during the manufacture of external and internal cylindrical steel parts with turning and grinding processes. Turning and grinding have been selected as the focus of the research due to their widespread use in manufacturing industries. Two types of turning process were considered: chucking and centre. In chuckingtype turning, a workpiece is held in a chuck or collet on a lathe spindle. In centretype turning, a workpiece is supported between work centres mounted on a spindle and the tailstock of a lathe. The external and internal cutting operations are assumed to be performed on an engine lathe with one single-point cutting. External and internal cylindrical grinding processes were also considered. Three popular methods are used for cylindrical grinding: centre, centreless and chucking. Centre-type grinding occurs when work centres are mounted on the headstock and tailstock of a grinder and the workpiece is rotated between the centres. This method is typically used to create single- and multiple-diameter shafts. For centreless grinding, a workpiece is not held physically in place while being ground. Instead, it rests on a workrest blade and is backed up by a regulating wheel. Chucking-type grinding is similar to chucking-type turning. However, instead of using a cutting tool, a grinding wheel in a grinder removes material. Chucking grinders are used for short shaft and hole grinding. In this research, typical form and size errors generated by these machining processes were studied; however, the proposed methods may also be applied to other machining methods. Many factors influence the form and size errors that are generated in turning and grinding operations, including the design of the machine tools and fixtures, cutting tool materials and their geometry, cutting parameters, thermal expansion, system vibration, system distortion, and machine maintenance condition. This research
Unified functional tolerancing for precision cylindrical components
27
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
focuses on the study of such typical factors (i.e. error sources) as spindle errors, cutting tool wearing, workpiece load deformation, thermal expansion, fixturing conditions and machine set-up. From a statistical perspective, the geometric errors produced in a batch of workpieces can be divided into two categories: systematic and random. A systematic error occurs when a trend of altitude and direction can be observed over time from workpiece to workpiece. For instance, the size error created by tool set-up on a batch of workpieces is constant, while the trend of normal wear on a cutting tool is linear. Systematic errors can often be controlled by a machine operator. In contrast, form errors created by variations in residual stress and in the cutting stock of a workpiece are regarded as random errors since errors from part to part are unpredictable. Random errors are normally uncontrollable. Thus, the geometric errors studied in this research were systematic.
2. Errors in turning and grinding processes The geometry of every manufactured workpiece deviates to some extent from that of the ideal (i.e. designed and intended) part geometry. In manufacturing, the geometric deviation of a cylindrical part can be described in three independent terms: cross-section form error, axial form error and cross-section size error. 2.1 Cross-section form errors For manufactured cylindrical parts, cross-section form error is defined as the error of out of roundness of the part, as measured in a cross-section perpendicular to the axis of that part. The following sections discuss models of cross-section form errors created by spindle defects and fixturing distortion. 2.1.1 Spindle rotation errors. The spindle of a lathe or a chucking-type grinder is supported by a series of bearings located in the headstock of the machine tool. Imperfections in the bearings can make the spindle unstable as it rotates in the radial direction. In turn, radial variations in spindle rotation cause the roundness of the part being manufactured to deviate from that of the ideal part. Typical imperfections in bearing components include geometric form and size errors, which are located in such bearing components as the ball bearing, the roller and the groove. Imperfections in these components often produce sine and cosine errors in the finished machined workpiece. The following example illustrates a periodical error induced in a finished workpiece by an oversized ball bearing in a lathe spindle (figure 1(a)). The ball bearing supports the spindle, and the workpiece is mounted on the spindle using a chuck. The outer groove of the bearing is fixed on the lathe, while the inner groove, spindle and workpiece rotate together. Let the inner groove of the ball bearing be C1, the oversized ball be C2 and the outer groove be C3 (figure 1(b)). Assume that the rotation speed of the inner groove is !0, and the speed of the outer groove is zero. The radii of the inner and outer grooves are R1 and R3, respectively; and the radius of the oversized ball bearing and the diameter error of the oversized ball are R2 and D2, respectively. The diameter error of a ball, D2, pushes the workpiece off centre. The movement of the centre
28
X. D. Zhang et al.
y C3 C2 C1
Workpiece
Cutter
(a)
x
(b) Figure 1.
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
0
Error of a spindle bearing.
of the workpiece, which occurs in the x direction, causes a roundness error on the workpiece surface, which in turn defines the error function of the surface with respect to the diameter error of the ball. 1 R1 !2 ¼ ! : 2 R2 0 The rotation speed (!2) of the ball with respect to C3 is expressed as: 2R2 !2 ¼ R1 !0 : The velocity at the centre of the ball (v2), relative to the centre of C1, is expressed as: 1 2 ¼ R1 !0 2 The rotation angle (2) of the centre of the ball from the x-axis at time t is: Zt 1 1 R1 2 ðtÞ ¼ dt ¼ ! t þ 2 ðt ¼ 0Þ R1 þ R2 0 2 2 R1 þ R2 0 When 2(t ¼ 0) ¼ 0, 2 ðtÞ ¼
1 R1 ! t: 2 R1 þ R2 0
The centre variation (r(t)) of the workpiece, which occurs in the x-axis direction due to the diameter error (D2) of the oversized ball, is: 1 R1 !0 t or D2 cos 2 R1 þ R2 1 R1 !0 t ð1Þ rðtÞ ¼ D2 cos½2 ðtÞ þ ¼ D2 cos 2 R1 þ R2 þ D2 1 R1 1 rð1 Þ D2 cos 2 R1 þ R2 This example demonstrates that an oversized ball causes spindle movement against the cutter, which can be measured as a cosine function of time. Because the cutter is fixed, the radius of the machined workpiece becomes non-uniform and varies as shown by the cosine function of the polar angle r(). Obviously, many other ball or inner groove defects of a journal bearing can lead to sine and cosine form errors
Unified functional tolerancing for precision cylindrical components
Clamping
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Figure 2.
Grinding
29
Releasing
Manufactured error by the clamping force.
on a machined cylindrical workpiece. If a dominant defect in the bearing components exists, a machined surface error may appear in an obvious sine pattern (e.g. a two-lobing shape, three-lobing shape, etc.). In general, as expressed in equation (2), the geometric error, r(), as created on the machined surface by the spindle bearing components, is a combination of the sine and cosine functions: rðÞ ¼ r0 þ rðÞ X X Ai cosðiÞ þ Bi sinðiÞ ¼ r0 þ
ð2Þ
where is the angle of cross-section, r the radius of a workpiece at , r0 the average radius of a workpiece, r the radius error of a workpiece at , and Ai, Bi are the constants. 2.1.2 Errors by fixturing distortion. For workpieces with low rigidity, fixturing forces can cause outward-round-roundness errors on a workpiece. For instance, in internal grinding, a three- or four-jaw chuck is normally applied to hold a workpiece. If a workpiece has low rigidity, the clamping force exerted by the three-jaw chuck distorts the shape before grinding (figure 2). After the workpiece is ground, the clamping force is released and the ground hole becomes a three-lobe shape. Similar phenomena occur with a four-jaw chuck, which creates a four-lobe shape. An approximate model of the error created by the three-jaw chuck is: rðÞ ¼ r0 þ rðÞ ¼ r0 þ A cosð3Þ
ð3Þ
where r0 is an average radius and A is a constant. Similarly, an approximate model of the error by the four-jaw chuck is: rðÞ ¼ r0 þ rðÞ ¼ r0 þ A cosð4Þ
ð4Þ
where r0 is an average radius and A is a constant. 2.2 Axial errors Axial errors of a manufactured cylindrical part are errors that occur in a section plane that passes through the axis of the manufactured part. Typical axial errors in cylindrical turning and grinding processes are discussed below. Some are related to machine tools (e.g. spindle misalignment), while some others are associated with workpieces (e.g. workpiece deflection and heat expansion). Still others deal with machine fixtures (e.g. work centre misalignment).
30
X. D. Zhang et al.
Workpiece rkpiece Work rk centre
Work centre r
Feeding eding direction Cutting tting tool Figure 3.
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
View
Work centre
Taper error created by horizontal misalignment.
Workpiece a
b Feeding direction Figure 4.
Work centre
Cutting tool
∆r a b Enlarged view
Hyperbolic error created by vertical misalignment.
2.2.1 Errors created by the misalignment of spindles and work centres. Spindle misalignment occurs when the spindle and work centres are not aligned to match the feeding direction of a cutting tool or the traverse direction of a grinding wheel. Misalignment causes cylindricity errors to occur along the axis of the piece. Spindle or work centre misalignment consists of two parallelism errors: one in a horizontal plane and one in a vertical plane. These create different geometric errors on a manufactured workpiece. If the misalignment between the axes of two work centres is offset in a horizontal plane, it creates a taper error of the same amount of the offset, r. This measure is approximately equal to the difference between the maximum and minimum radii (figure 3). The taper error model is shown in equation (5): rðzÞ ¼ r0 þ rðzÞ ¼ r0 þ Az
ð5Þ
where z is the axis of the workpiece, r0 is an average radius and A is a constant. When work centres are misaligned in the vertical plane, the geometric error of the manufactured workpiece is hyperbolic in a cross-sectional plane that passes through the workpiece axis, since the cutting trace is at an angle to the axis of the workpiece (figure 4). A projected view can be used to calculate the form error, r. The enlarged view in figure 4 is a projection of the workpiece on a plane perpendicular to the workpiece axis. In the enlarged view, the outer circle is the diameter at the workpiece end face, while the inner circle is a cross-section diameter in the middle of the part. The form error, r, is determined via the misalignment and the workpiece diameter. Note that the vertical distance, ‘, of ab in the enlarged view is approximately equal to the misalignment of the work centres. Assuming that the average diameter of the workpiece is D, the form error, r, is calculated using equation (6). Since the vertical distance, ‘, is normally much smaller than the diameter, D, r
Unified functional tolerancing for precision cylindrical components
31
is much smaller than ‘ and can be ignored in practice: ‘2 ¼ rðD rÞ 4
ð6Þ
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
‘2 r 4D 2.2.2 Errors caused by workpiece deflection. When a shaft is turned or ground in a chuck or between work centres, the cutting force generates workpiece deflection (figures 6 and 8). Deflection makes a workpiece move away from the cutting tool. The degree of the deflection at the cutting location (point) varies, depending on how far the cutting tool is from the supporters, such as the chuck and work centres. The further the cutting tool is from the supporters, the greater the deflection at the cutting location. Thus, the deflection at the cutting location varies along the axis of the machining part and is a function of the distance of the tool from the supporter. Variation of the deflection at the cutting location results in a geometric form error on the machined workpiece. The form error is equal to the total variation of the deflection at the cutting location. The pattern of the error is determined by the deflection function. Deflection can be analysed using material mechanics, which makes this matter an issue of beam deflection calculation (Green 1992, Boresi et al. 1993). Two common cases in turning and grinding processes were studied. In the first case, a shaft is held by a chuck. According to mechanics of materials, this case can be treated as a beam fixed at one end under an intermediate load (figures 5 and 6). The deflection, y, at the cutting position (Green 1992) is: y¼
Wx3 6EI
ð7Þ
Chuck Workpiece
Cutting tool Figure 5.
Workpiece in a chuck.
y l
x x W Figure 6.
Beam fixed at one end.
32
X. D. Zhang et al.
where E is the modulus of elasticity of the material, I the moment of inertial of the cross-section of the beam, W the load on beam, and y the deflection of the cutting location. The form error caused by the deflection can be rewritten as polynomial functions, as shown in equation (8). If Legendre functions are used to model the form error, equation (8) can approximately represent equation (9). In equation (9), the taper and parabolic functions replace the cubic power function: rðzÞ ¼ r0 þ rðzÞ
ð8Þ
¼ r0 þ Az3
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
rðzÞ ¼ r0 þ rðzÞ ¼ r0 þ Bz þ Cð3z2 1Þ=2
ð9Þ
where z is the coordinate of workpiece axis, r the radius of a workpiece at z, r0 average radius of a workpiece, r the radius error of a workpiece at z, and A, B, C are the constants. The second case is one in which a shaft is held between work centres. This creates a problem wherein a beam is supported at both ends under an intermediate load (figures 7 and 8). Using material mechanics, the deflection, y, at the cutting position is (Green 1992): y¼
W 2 2 ‘ x 2‘x3 þ x4 3EI‘
ð10Þ
where E is the modulus of elasticity of the material, I the moment of inertial of the cross-section of the beam, W the load on beam, and y the deflection of the cutting location.
Work centre
Work centre Workpiece
Cutting tool Figure 7.
Workpiece between work centres.
y l
x x W Figure 8.
Beam supported by both ends.
Unified functional tolerancing for precision cylindrical components
33
Expressing the deflection in terms of polynomial functions, the form error is rewritten as equation (11). The approximate model in Legendre functions is shown in equation (12): rðzÞ ¼ r0 þ rðzÞ ¼ r0 þ Az2 þ Bz3 þ Cz4 rðzÞ ¼ r0 þ rðzÞ ¼ r0 þ Dð3z2 1Þ=2
ð11Þ
ð12Þ
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
where z is the coordinate of workpiece axis, r the radius of a workpiece at z, r0 the average radius of a workpiece, r the radius error of a workpiece at z, and A, B, C, and D are the constants. 2.2.3 Errors caused by thermal expansion of the workpiece. In turning and grinding operations, the energy dissipated in cutting operations is converted into heat, which in turn raises the temperature of the cutting zone. The temperature increases as the strength of workpiece material, cutting speed and depth of the cut increases (Boresi et al. 1993). This temperature increase can cause expansion in the size of the workpiece. During a cutting operation, the gradual change in temperature generates geometric form errors in the final workpiece. Using general physics, the thermal expansion of workpiece diameter can be calculated using equation (13) (Green 1992). Since the temperature increase can be expressed as a linear function of time (Boresi et al. 1993), the form error caused by the heat is a taper as expressed in equation (14): D ¼ CDðT T0 Þ ð13Þ where D is the change in diameter, D the diameter, C the thermal conductivity, T0 the initial temperature of a workpiece, and T the final temperature of a workpiece. rðzÞ ¼ r0 þ rðzÞ
ð14Þ
¼ r0 þ Az where z is the coordinate of the workpiece axis, r the radius of a workpiece at z, r0 the average radius of a workpiece, r the radius error of a workpiece at z, and A the constant. 2.3 Cross-section size errors For this research, cross-section size errors of cylindrical workpieces were measured with respect to the diameter of the workpiece. The diameter of a manufactured cylinder is defined as an average diameter of the manufactured workpiece. Size (diameter) error of a manufactured cylindrical part is equal to the difference between the actual and the predicted size (diameter or radius) of the manufactured part. In precision turning and grinding processes, many factors affect part size, the most common of which are set-up error and tool wear. Diameter error caused by these two factors can be written in the form of equation (15): rðzÞ ¼ r0 þ rðzÞ ¼ r0 þ A
ð15Þ
34
X. D. Zhang et al.
where z is the coordinate of the workpiece axis, r the radius of a workpiece at z, r0 the average radius of a workpiece, r the radius error of a workpiece at z, and A the constant.
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
3. Modelling of geometric errors with L–F polynomials The above sections show that geometric form and size errors created in the manufactured cylindrical workpieces by individual manufacturing error sources can be modelled using L–F functions (Beckmann 1978, Glenn 1984). L–F functions provide a guideline for manufacturing engineers to identify the sources of systematic manufactured errors in the manufacturing process. The identified errors can then be removed so that the manufacturing process becomes under control. Therefore, L–F polynomial models provide a bridge between manufactured geometric errors and sources of those errors. After a workpiece is manufactured by a process operation, such as turning and grinding, it is measured by an inspection device, such as a coordinate measuring machine (CMM), to obtain a data set {r, , z} for the geometry of the manufactured workpiece. The geometry in the data set {r, , z} can be mathematically decomposed into L–F polynomial functions, as shown in equation (16) (Glenn 1984). Each coefficient of the polynomials represents a cross-section form error, axial form error or cross-section size error: rð, zÞ ¼ r0 þ rð, zÞ XX X ½Aij Pj ðzÞ cosðiÞ þ Bij Pj ðzÞ sinðiÞ ¼ A0j Pj ðzÞ þ
ð16Þ
where r is the radius of a workpiece at (, z), r0 the average radius of a workpiece, r the form error of a workpiece at z, P the Legendre function, and A, B are the constants. Since the L–F polynomial functions are orthogonal, their coefficients (Aij, Bij) can be calculated using equations (17) and (18) (Beckmann 1978): Pt Pu rij Pj ðzk Þ cosðil Þ P Pl¼12 Aij ¼ k¼1 ð17Þ Pj ðzk Þ cos2 ðil Þ Pt Bij ¼
Pu rij Pj ðzk Þ sinðil Þ P Pl¼12 Pj ðzk Þ cos2 ðil Þ
k¼1
ð18Þ
where l is the order of an azimuthal angle, u the total number of azimuthal angles, k the order of a point in z-axis, and t the total number of points in z-axis. Table 1 lists major manufactured errors for cylindrical parts and their L–F polynomials representations; table 2 depicts four of the typical errors graphically. Note that different errors may be represented by the same L–F polynomial model parameter, such as the diameter errors caused by both tool set-up and heat expansion of workpiece can be represented by A00 since they belong to the same type of error. As discussed above, typical errors on a manufactured cylindrical workpiece occur in lobes in the cross-section; in taper and in convex or concave shapes in the axial
Unified functional tolerancing for precision cylindrical components Table 1.
35
Typical manufactured errors and L–F polynomials representations.
Manufactured error and source
L–F polynomials model
Cross-section error by spindle rotation error S[Ai cos(i) þ Bi sin(i)] Cross-section error by fixturing distortion A30 cos(3) or A40 cos(4) Axial error by misalignment of spindle A01z Axial error by misalignment of work centres A01z Axial error by deflection of workpiece hold in a chuck A01z þ A02 (3 z2 1)/2 Axial error by deflection of workpiece hold between work centres A02 (3 z2 1)/2 Diameter error by tool set-up A00 Diameter error by heat expansion of workpiece A00
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Aij, constant; , angular coordinate of cross-section; z, coordinate of workpiece axis.
Table 2.
Representation and F–L polynomials models of four typical manufactured errors.
Taper
A00 þ A01z
Concave
Convex
Banana
A00 þ A02(3z21)/2
A00–A02(3z2 1)/2
A00 þ A12((3z2 1)/2)cos
Table 3.
Definitions of the proposed tolerance parameters.
Tolerance character
Symbol
Value
Diameter Lobing
1 IL
Positive taper
2 2 s s
2A00 2Ai0 or 2Bi0 (i > 1) 4A01
Negative taper Concave Convex
4A01 1.5A01 1.5A01
Definition Average diameter Maximum lobing variation of the i lobing shape Maximum diametric difference of a cylinder where the diameter is larger with the increase of z Maximum diametric difference of a cylinder where the diameter is smaller with the increase of z Maximum difference along z-axis where the radius at z ¼ 0 is smallest on the cylinder Maximum difference along z-axis where the radius at z ¼ 0 is largest on the cylinder
section; and in diametric variation in the cylindrical section. Thus, tolerance specification was studied in this research only for these typical tolerances, even though the methodology can be applied to other types of the manufactured errors. Table 3 shows the definitions of the proposed tolerance parameters based on L–F function coefficients. In practice, the proposed tolerance parameters complement the standard ‘peak-to-valley’ tolerance parameters generated in the total profile of a manufactured part. Standard tolerance parameters may provide a general picture of the dimensional variations that occur in the parts. The proposed tolerance parameters provide details of manufactured errors, and thus offer an effective tool for error
36
X. D. Zhang et al.
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
diagnosis and correction/control. Since dominant components of a total manufactured error can be modelled by limited lower order F–L functions, the computational complexity of the modelling process is low. In design and manufacturing, the proposed tolerancing scheme can be used as follows: . Based on the part functions, a set of the new geometric tolerances is specified. . Part is measured with a CMM and the part geometry data (coordinates of measurement points) are obtained. . Tolerance parameters for the specified geometric errors are calculated using the L–F polynomial function fitting with the measured data. . Calculated geometric errors are checked against the specified tolerances and decision on the part quality is made. When the errors are out of the specified tolerances, the errors can be diagnosed and corresponding adjustments can be made to the machine to eliminate the geometric errors.
4. Functional tolerancing with the proposed tolerancing scheme In design and manufacturing practice, to ensure that a manufactured product satisfies design functionality, designers specify the geometric limits in terms of the geometric dimensions and tolerances on the design drawings for manufacturing production. Although the basic objective of dimensioning and tolerancing is to create a mechanical system that meets functional requirements, the route to achieve this goal is usually indirect. Designers have to observe a number of standards and technical rules related to mechanical, dimensional and geometric tolerances to produce a design which is consistent with not only the functions of the part, but also the manufacturing constraints, assembly requirements and inspection feasibility. These latter conditions essentially determine the economy of the component manufacturing. Therefore, a primary concern of the design and manufacturing functions in industry is to strive for a tolerancing system that would support designers in finding an optimum solution from functional requirements and manufacturing constraints to a comprehensive tolerance design of mechanical parts. Tolerance information is essential for designing, process planning, manufacturing, assembly, inspecting, and many other production activities. Geometric dimensioning and tolerancing plays an imperative role as a ‘medial’ among design, manufacturing and metrology. A number of efforts have been made for functional tolerancing. In a study of mechanical rolling and sliding functions, Whitehouse (1994) investigated the relationship of contact stress and behaviour of geometric features. He concluded that the peak characteristics of a surface are the only feature of interest for the rolling and sliding functions. Hoeprich (1993) investigated geometric variation effects on rolling bearing fatigue life. Bhargava and Grant (1993) studied the correlation of crankshaft bearing wearing failure of diesel engine and manufacturing waviness. In a study of scuffing and seizing in a fuel injector (Ludema and Bai 1996), the geometry variation errors of roundness and straightness were found to be major factors to the scuffing and seizing. Srinivasan (1994) and Srinivasan et al. (1996) proposed the fractal-based tolerance parameters for waviness variation control of mechanical products. Marguet and Mathieu (1997) found the aircraft
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Unified functional tolerancing for precision cylindrical components
37
assembly function closely relied on the geometric tolerances. O’Connor and Spedding (1992) conducted a study that aimed to establish the surface conditions necessary to obtain the desired functional performance. Jackman et al. (1994) developed a method for assessing the compliance of several cylindrical part features based on comparing tolerance specifications with actual measurement data. Although some progress has been made in this field, more investigations are needed for developing and implementing systematic functional dimensioning and tolerancing approaches. The present research conducted a study on the relevance of form and diameter errors on fuel leakage. The problems with traditional tolerance parameters defined in current standards were addressed and investigated for the purpose of leakage variation control of the fuel injectors. By using the proposed L–F polynomials tolerance parameters, the functional tolerancing for fuel injector leakage variation control was investigated. The proposed tolerance parameters were compared with traditional ones from a viewpoint of the leakage function relevance. The study of the leakage relevance was carried out using the data from computational fluid dynamics (CFD) simulation. CFD has been proven to be an effective modelling technique for predicting fluid dynamics behaviour and has been widely used in industry. Finally, the functional tolerancing approach was illustrated by an example of tolerancing of fuel injector components for leakage variation control.
4.1 Functional tolerancing Various mechanical products serve many different functions, such as energy conversion, motion transmission and control. These functions can be realized by specified physical, mechanical and geometrical properties of parts. Due to the limitations on materials and the precision of manufacturing and assembly processes, variations always exist in product properties, which in turn, result in product function variations. Among the product properties, geometric parameters, which include dimension, form and surface quality, are considered key factors in controlling product function variations. To control the variations on product functions, proper tolerances need to be specified on geometric parameters. This procedure is called functional tolerancing. The result of this practice has not been satisfactory in the manufacturing industry due to the following deficiencies of the tolerance parameters defined in current standards: . Lack of functional relevance: single valued, maximum peak-to-valley form tolerances, such as straightness and roundness, do not demonstrate a strong correlation with product functions in many cases, although they may be good parameters for assembly fit function. . Lack of manufacturing relevance: maximum peak-to-valley form tolerances do not have a strong link to the problems in manufacturing processes and therefore, are difficult to control in manufacturing. In Section 3, a set of L–F polynomials based tolerance parameters was introduced. These new tolerance parameters were shown to have strong relevance to manufacturing controllability. In the following sections, the relevance of the new tolerance parameters to part functions was investigated, and specifically, a case study
38
X. D. Zhang et al.
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Figure 9.
Fuel injector components.
of functional tolerancing for the control of fuel leakage variation in fuel injectors of a diesel engine is presented. In the fuel systems of diesel engines, applying high pressure is a common method to improve fuel economy and reduce emissions. To achieve a pressure of over 20 000 psi, the fuel leakage between injector components must be tightly controlled since a small leakage variation between injector components can substantially affect the fuel injection pressure. Consequently, the pressure variation will lead to poor performance (i.e. lower reliability and greater emission of the engines). Thus, the sliding mating components (cup and needle) of a fuel injector (figure 9) must fit each other tightly to prevent fuel leakage under high pressures. Further, the contact force between the cup and needle must be small to prevent the sliding pair from seizing; therefore, the gap is strictly controlled in tight dimensional and geometric tolerances. In current practice, designers assign tight geometric tolerances (roundness, straightness and taper) to the cup and needle, where diameter tolerance is in the same scale as the form tolerances. The standards define form tolerances only in terms of a maximum peak-to-valley value of a form deviation from perfect form. Industrial experience proves that those parameters do not relate the sealing function well. Therefore, they lead to inconsistent results in the fuel leakage control.
4.2 Functional tolerancing of fuel injector components The issue of tolerancing a manufactured part was addressed by investigating the sources and causes of manufactured form errors. A good tolerancing method requires that the tolerance parameters cannot only represent the manufactured form errors but also correlate the sources in a manufacturing system to the manufactured errors. For every manufactured part, some geometric deviation from ideal exists. The geometric error of a cylindrical manufactured part can normally be decomposed into three components: form error in cross-section, form error in axial direction and size error (Zhang et al. 1999, Zhang 2001). These errors may be caused by the imperfections in machine tool, fixture, workpiece, cutting process, etc., as analysed in Section 2. These manufactured errors can be expressed by the L–F polynomials.
Unified functional tolerancing for precision cylindrical components
39
For the example part, the internal surface of the cup and the external surface of the needle are manufactured by the processes of tuning and grinding. These errors can be well represented by the L–F polynomials.
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
4.2.1 Influence of manufactured form errors on fuel leakage variation. Form errors on manufactured needle and cup (figure 9) of a fuel injector may have a different affect the fuel leakage between the needle and cup. This research focused on the difference in the influence of the different form errors. The errors for the study were divided into two categories, cross-section form error and axial form error. The influence of the form errors to the leakage variation was studied through the CFD simulations. 4.2.2 CFD simulation conditions and terminology. CFD simulations were set for the conditions of practical fuel injectors in a diesel engine. However, without loss of generality, some assumptions were made in the CFD simulations. The internal cylindrical surface of the cup had geometric errors in form and size, while the external cylindrical surface of the needle was fixed in that it had no form error. The diameter of the external cylindrical surface of the needle was 10 mm. The match length of the needle and the cup was 12 mm. The fluid of the No. 2 diesel was applied in the simulation. The viscosity was 0.008 kg/m-s; the fluid temperature at inlet was 273.15 K (or 80 C); the fluid density was 905 k/m3; the specific heat of fluid was 2105 J/kg-K; and the thermal conductivity of fluid was 0.149 W/m-K. The flow was laminar, steady and fully developed. The inlet pressure was 20 000 psi and the outlet pressure was zero. In the simulations, the structural deformations on the needle and cup were not considered. The relative movement or velocity between the needle and cup was zero. The CFD code of FLUENT 5.3 was employed. For the simulation study, certain terms should be defined: Mating pair Match clearance
Pair of mating hole (cup) and shaft (needle). Average radial distance between hole and shaft of a mating pair. Perfect mating pair Mating pair in which the hole and shaft have perfect geometry form. Corresponding perfect Perfect mating pair in which the hole and shaft have perfect mating pair of a surface in geometry and the equal average clearance of the mating pair mating pair. Leakage, L Fuel leakage between the clearance hole and shaft of a mating pair. Base leakage, L0 Leakage of a corresponding perfect mating pair. Relative leakage Relative difference in the fuel leakage of a mating pair and variation its corresponding perfect mating pair. Normalized leakage Ratio of the relative leakage variation to the base leakage variation rate (NLVR) of a mating pair: NLVR ¼ (L L0)/L0(%). Roundness form error Maximum radial variation on the edge profile of an axisperpendicular cross-section of a part of a mating pair. Lobe number Frequency of a sine or cosine form of the roundness error. Straightness form error Maximum radial variation on the edge profile of a passing-axis cross-section of a part of a mating pair.
40
X. D. Zhang et al.
Convex shape Concave shape Taper form error Positive taper shape
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Negative taper shape
Parabolic form of straightness form error, which appears higher in the middle than a part of a mating pair. Parabolic form of straightness form error, which appears lower in the middle than a part of a mating pair. Maximum diametric error on a conical surface of a part of a mating pair. Form with a smaller diameter at an inlet of a conical surface of a part of a mating pair. Form with a larger diameter at the inlet of a conical surface of a part of a mating pair.
4.2.3 Cross-section profile error versus leakage variation. Since a cylindrical surface is normally formed by the manufacturing methods of turning, drilling and grinding, manufactured geometric errors on a cross-section of the surface will likely produce a periodic error or a combination of periodic errors. Using the Fourier functions, a manufactured cross-section profile can be decomposed into a series of sine and cosine functions (figure 10). The relevance of cross-section profile errors to the leakage was studied using the CFD simulation. In the cross-section profile errors study, the factors of match clearance, amplitude, and lobing were investigated. In table 4, the factor of match clearance to the leakage variation was studied by fixing the factors of roundness error and lobing. The simulation results indicated that a cross-section form error in a smaller match clearance increased the leakage by 9.5% in a term of NLVR. In Cases 3–5 in table 5, the authors studied the factors of roundness error amplitude by fixing match clearance and lobing, resulting in the larger roundness error causing more leakage by 9.5% (table 5 and figure 11). In Cases 6 and 7 in table 6, the factor of lobing was investigated, resulting in no significant influence to the leakage.
+
=
+ ...
+
+
Figure 10. Decomposition of a manufactured cross-section profile. Table 4. Case 1 2
Match clearance (mm)
Roundness error (mm)
Roundness lobe
Leakage (kg/s) e-5
Base leakage (kg/s) e-5
NLVR (%)
2.0 3.0
1.0 1.0
2 2
3.0379 9.7652
2.7753 9.3788
9.5 4.1
Table 5. Case 3 4 5
Influence of match clearance.
Influence of roundness error amplitude.
Match clearance (mm)
Roundness error (mm)
Roundness lobe
Leakage (kg/s) e-5
NLVR (%)
2.0 2.0 2.0
0.0 0.5 1.0
2 2 2
2.7753 2.8427 3.0379
0.0 2.4 9.5
41
Unified functional tolerancing for precision cylindrical components Leakage variation rate by roundness error
NLVR
10.0% 9.0% 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Roundness error (µm)
Figure 11. Influence of roundness error amplitude. Table 6. Case
Match clearance (mm)
Roundness error (mm)
Roundness lobe
Leakage (kg/s) e-5
2.0 2.0
1.0 1.0
2 4
3.0379 3.0377
6 7
Table 7.
Case 8 9 10 11 12 13 14 15 16 17
Influence of roundness lobing.
Influence of positive taper shape with various average match clearances.
Match clearance (mm)
Positive taper (mm)
Leakage (kg/s) e-5
NLVR (%)
2.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 3.0
0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
2.7753 2.7717 2.7555 2.7286 2.6909 9.3788 9.3657 9.3410 9.3012 9.2438
0.0 0.1 0.7 1.7 3.0 0.0 0.1 0.4 0.8 1.4
4.2.4 Axial profile error versus leakage variation. In the axial direction of a manufactured surface, a profile error can be expressed in the Legendre polynomial functions. In this study of the leakage relevance, four typical manufactured profile errors (positive taper, negative taper, convex and concave) were investigated for the leakage variation impact. CFD simulation was used to calculate the leakage. For the leakage study, the mating parts chosen represented a variety of amplitudes and undulations of profile errors, and clearances between mating parts. In Cases 8–17 of the positive tapers, the results (table 7 and figure 12) showed the leakage tended to be smaller as the taper error was larger. The taper influence on the leakage became stronger as the
42
X. D. Zhang et al.
Leakage variation rate by taper form error 0.0% 0.00 -0.5%
0.20
0.40
0.60
0.80
1.00
1.20
NLVR
-1.0% -1.5%
3 um clearance
-2.0% -2.5% -3.0% 2 um clearance
-3.5%
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Positive taper (mm ) Figure 12. Influence of taper shape with difference average match clearances.
Table 8.
Leakage comparison of negative taper and positive taper. Negative taper error (mm)
Match clearance (mm)
Leakage (kg/s) e-5
NLVR (%)
18 19 20 21 22
0.00 0.25 0.50 0.75 1.00
2.0 2.0 2.0 2.0 2.0
2.7753 2.7716 2.7560 2.7285 2.6908
0.0 0.1 0.7 1.7 3.0
23 24 25 26 27
Positive taper error (mm) 0.00 0.25 0.50 0.75 1.00
2.0 2.0 2.0 2.0 2.0
2.7753 2.7717 2.7555 2.7286 2.6909
0.0 0.1 0.7 1.7 3.0
Case
match clearance was smaller; however, the decrease in leakage was not significant. For example, when the clearance was 2 mm, the leakage at a 1 mm taper only decreased by 3%. Table 8 shows the results of the study of the leakage difference of negative tapers from positive tapers. The influence by a negative taper error was the same as a positive taper error’s influence. The results in table 9 and figure 13 demonstrate that the leakage became smaller as the convex form error was larger, but the decrease in leakage was comparably large. When the clearance was 2 mm, the leakage change by the 1 mm convex error decreased by 17%. The influence of a convex form error on the leakage became stronger as a match clearance was smaller. In a case of concave shape, table 10 and figure 14 show that the leakage by the concave shape error was smaller than the one by the convex shape error. As shown in table 11 and figure 15, a combined form error leads to more leakage reduction. In the case where match clearance was 2 mm, the taper was 1 mm and the convex form error was 1 mm. The NLVR was about 23%.
Unified functional tolerancing for precision cylindrical components Table 9.
Case
Convex form error (mm)
Leakage (kg/s) e-5
NLVR (%)
2.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 3.0
0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
2.7753 2.7373 2.6619 2.5109 2.3005 9.3788 9.3047 9.1959 8.9652 8.6466
0.0 1.4 4.1 9.5 17.1 0.0 0.8 2.0 4.4 7.8
Leakage variation by convex form error 0.0% -2.0%0.00
0.20
0.40
0.60
0.80
1.00
1.20
-4.0% -6.0% NLVR
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Influence of convex shape with various average match clearances.
Match clearance (mm)
28 29 30 31 32 33 34 35 36 37
43
-8.0%
3 um clearance
-10.0% -12.0% -14.0% -16.0% -18.0%
2 um clearance
-20.0% Convex form error (mm)
Figure 13. Leakage difference of convex shape with various average match clearances. Table 10. Leakage influence of concave shape. Case 38 39 40 41 42
Match clearance (mm)
Concave form error (mm)
Leakage (kg/s) e-5
NLVR (%)
2.0 2.0 2.0 2.0 2.0
0.00 0.25 0.50 0.75 1.00
2.7753 2.7602 2.6983 2.6041 2.4767
0.0 0.5 2.8 6.2 10.8
4.3 Tolerance characterization using L–F functions As discussed in Section 2, typical errors on a manufactured cylindrical workpiece occur in lobes in the cross-section, in taper and in convex or concave shapes in the axial section. These manufactured errors have a different impact on the leakage. Thus, this research only performed tolerance characterization for these typical manufactured errors, even though the methodology can be applied to other types
44
X. D. Zhang et al. Different form errors vs leakage variation rate at 2 µm match clearance
0% -2%0.00
0.20
0.40
0.60
0.80
1.00
-4%
1.20
taper
NLVR
-6% -8% -10% -12%
concave
-14% -16%
convex
-18%
Figure 14.
Comparison in the leakage inference by different shapes. Table 11. Leakage by combined form error.
Case Test Test Test Test Test Test
Match clearance (mm)
Positive taper error (mm)
Concave form error (mm)
Leakage (kg/s)e-5
NLVR (%)
2.0 2.0 2.0 2.0 2.0 2.0
1.00 1.00 1.00 1.00 1.00 1.00
0.0 0.5 1.0 0.0 0.5 1.0
2.7753 2.6364 2.4432 2.7753 2.5395 2.1390
0.0 5.0 12.0 0.0 8.5 22.9
43 44 45 46 47 48
Combined form errors vs leakage variation rate at 2 mm match clearance and 1 mm positive taper 0.0% 0
0.2
0.4
0.6
0.8
1
1.2
-5.0%
NLVR
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
Form error (mm)
-10.0%
Taper & concave
-15.0% -20.0% -25.0%
Taper & convex Form error (mm)
Figure 15. Comparison in the leakage inference by different combinations of form errors.
of the manufactured errors. Table 3 shows the definitions and calculations of the proposed tolerance parameters based on L–F function coefficients. In design and manufacturing practice, the proposed tolerance parameters do not replace but complement the standard ‘maximum peak-to-valley’ tolerance parameters generated in the total profile of a manufactured part. The standard tolerance parameters may
Unified functional tolerancing for precision cylindrical components
45
provide an approximate picture of the dimensional variations that occur in the parts. Proposed tolerances can provide the details of manufactured errors, and thus offer an effective tool for error diagnosis and correction/control.
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
5. Example The following example illustrates an industrial application of the proposed tolerances and their applications to functional tolerancing. The example part is a needle of a diesel engine fuel injector. The external cylindrical surface was machined using the centreless grinding process. To control the leakage variation rate (NLVR) in a 2-mm match clearance under 5%, the tolerances for the manufactured errors were specified as shown in figure 16. The needle was measured with the Mahr Federal MFU8 cylindricity machine. Figures 17 and 18 show the cross-section and axial traces of two the needles. Equations (2) and (3) were applied to calculate the tolerance parameters. For Part No. 1, the results (table 12) indicated that the concave shape exceeded its tolerance specification, and the deviation might be caused
10+/-0.002
0.001 0.001 +/- 0.0008 0.0008 -0.0004
Figure 16. Tolerancing a needle of a fuel injector.
Figure 17. Cross-section and axial traces of Part No. 1.
Figure 18. Cross-section and axial traces of Part No. 2.
46
X. D. Zhang et al. Table 12. Measurement results.
Characteristic Tolerance (mm) Part No. 1 Part No. 2
Roundness
Straightness
Taper
Concave
Convex
NLVR (%)
0.0007 0.0007 0.0005
0.001 0.00095 0.00087
0.0008 0.0002 0.0001
0.0008 0.0009
0.0004
þ /5 13 17
0.0010
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
by a spindle alignment error of a machine tool. The concave shape with the taper error resulted in the leakage variation rate exceeding the leakage variation. In the case of Part No. 2, the amount of NLVR was even larger compared with Part No. 1’s, which was mainly caused by its large convex shape.
6. Conclusions A new L–F polynomial-based tolerancing method was proposed. Typical geometric errors for cylindrical component manufacturing were modelled using this tolerancing method. This new tolerancing scheme was illustrated with an example of tolerancing of a diesel engine fuel injector component to control leakage. The conclusions from the study are listed below: . Geometric error caused by an individual manufacturing source normally appears in the shape of one or more functions of the L–F polynomials. . When the total geometric error of a manufactured part results from several error sources in a manufacturing system, the total error of the part can be decomposed into a series of L–F functions. Each L–F function will normally correspond to an error source within the manufacturing system. . Dominated manufactured error components can be identified through a comparison of the coefficients of L–F functions; thus, major error sources can be identified or estimated using all the error sources in the manufacturing system. The L–F polynomial modelling of a manufactured error can provide a convenient tool for manufacturing process control. . In the study of a cross-section profile error, the amplitude of the profile error is a moderate factor to the leakage variation. A cross-section profile error, regardless of its frequency, always makes the leakage larger in contrast to the controversy influence of an axial form error. In the case of the 2-mm match clearance, the leakage can be increased by 10%. . Axial profile errors such as positive and negative taper, and concave and convex profile errors always reduce the leakage, which is different from the inference of a cross-form error. When the match clearance is 2 mm, a 1 mm convex profile error can lead to a 17% leakage variation rate, while the inference of 1 mm concave form error is only 11%. The traditional tolerancing methodology fails to distinguish the difference. . Traditional tolerance parameters of the maximum peak-to-valley cannot effectively control leakage variation. The L–F polynomial-based tolerance parameters are a better means to bridge the manufactured errors to the leakage variation rate of a fuel injector, and thus the leakage control can be more effectively controlled.
Unified functional tolerancing for precision cylindrical components
47
Downloaded by [Georgia Tech Library] at 14:41 26 June 2014
References Asme, ASME Y14.5M-1994: Dimensioning and Tolerancing, 1994 (American Society of Mechanical Engineers: New York). Beckmann, P., Orthogonal Polynomials for Engineers and Physicists, 1978 (Golem: Boulder). Bhargava, S. and Grant, M.B., 1993, Taguchi method and tolerance assignment: the wavelength aspects of surface metrology, in Proceedings of the 1993 International Forum on Dimensional Tolerancing and Metrology, ASME, June 1993, pp. 177–182. Boresi, A.P., Schmidt, R.J. and Sidebottom, O.M., Advanced Mechanics of Materials, 1993 (Wiley: New York). Glenn, P., Set of orthogonal surface error descriptors for near-cylinder optics. Optics Engineering, 1984, 23, 384–390. Green, B.E., Machinery’s Handbook, 24th edn, 1992 (Industrial Press: New York). Hoeprich, M., Geometric variation effects on rolling element bearing life, in Proceedings of the 1993 International Forum on Dimensional Tolerancing and Metrology, ASME, June 1993, pp. 167–175. ISO, ISO 1101:1983: Technical Drawings—Geometrical Tolerancing—Tolerancing of Form, Orientation Location and Run-out—Generalities, Definitions, Symbols, Indications on Drawings, Tolerances of Form and of Position, 1983 (International Organization of Standardization: Geneva). Jackman, J., Deng, J.J., Ahn, H., Kuo, W. and Vardeman, S., Compliance measure for the alignment of cylindrical part features. IIE Transactions, 1994, 26, 2–10. Ludema, K. and Bai, D., Plunger scuffing investigation. Technical Report, 1996 (Cummins Engine Co.: Columbus). Marguet, B. and Mathieu, L., Tolerancing problems for aircraft industries, in Proceedings of the 5th CIRP International Seminar on Computer-Aided Tolerancing, Toronto, Canada, April 1997, pp. 335–343. O’Connor, R.F. and Spedding, T.A., Use of a complete surface profile description to investigate the cause and effect of surface features. International Journal of Machine Tools and Manufacture, 1992, 32, 147–154. Srinivasan, R.S., A theoretical framework for functional from tolerances in design for manufacturing. Ph.D. dissertation, University of Texas at Austin, 1994. Srinivasan, R.S., Wood, K.L. and McAdams, D.A., Functional tolerancing: a design for manufacturing methodology. Research in Engineering Design, 1996, 2, 99–115. Whitehouse, D.J., Handbook of Surface Metrology, 1994 (Institute of Physics Publishing: Bristol). Zhang, X.D., A unified functional tolerancing approach for integrated design, manufacturing, and inspection. Ph.D. dissertation, Florida State University, Tallahassee, 2001. Zhang, X.D., Zhang, C. and Wang, B., Modeling of geometric errors of manufactured parts for manufacturing process control, in Proceedings of the Fourteenth Annual Meeting of ASPE, 31 October–5 November 1999, pp. 146–149.
