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A Scheme for Mapping Tolerance Specifications to Generalized Deviation Space for Use in Tolerance Synthesis and Analysis Haoyu Wang, Nilmani Pramanik, Utpal Roy, Rachuri Sudarsan, Ram D. Sriram, Senior Member, IEEE, and Kevin W. Lyons
Abstract—Tolerances impose restrictions on the possible deviations of features from their nominal sizes/shapes. These variations of size/shape could be thought of as deviations of a set of generalized coordinates defined at some convenient point on a feature. Any tolerance specification for a feature imposes some kind of restrictions or constraints on its deviation parameters. These constraints, in general, define a bounded region in the deviation space. In this paper, a method has been presented for converting tolerance specifications as per maximum material condition (MMC)/least material condition (LMC)/regardless of feature size (RFS) material conditions for standard mating features (planar, cylindrical, and spherical) into a set of inequalities in a deviation space. Both virtual condition boundaries1 (VCB) and tolerance zones are utilized for these mappings. The mapping procedures have been illustrated with an example. Note to Practitioners—This paper deals with methods to convert tolerance specification as per ASME Y14.5M into a set of generalized deviation of features and vice-versa. These are intermediate relationships that are required for use in deviation-based tolerance synthesis methods. In this work, different examples have been presented to show how different tolerance specifications (such as positional tolerance at maximum material condition (MMC), least material condition (LMC), etc.) applied to different features could be treated on a generalized basis for tolerancing of manufactured parts. Index Terms—Deviation space, mapping, tolerance analysis, tolerance synthesis. Manuscript received April 14, 2005. This paper was recommended for publication by Editor M. Wang upon evaluation of the reviewers’ comments. A condensed version of this paper was presented in ASME 2002 Design Engineering Technical Conference and Computers and Information in Engineering Conference, 28th Design Automation Conference, Sept. 29–Oct 2, 2002, Montreal, QC, Canada. H. Wang is with the Department of Applied Sciences, Bowling Green State University/Firelands, Huron, OH 44839 USA (e-mail:
[email protected]). N. Pramanik is with the Department of Industrial Technology, University of Northern Iowa, Cedar Falls, IA 50614 USA (e-mail:
[email protected]). U. Roy is with the Department of Mechanical, Aerospace and Manufacturing Engineering, Syracuse University, Syracuse, NY 13244 USA (e-mail:
[email protected]). R. Sudarsan is with Engineering Management and Systems Engineering, George Washington University, Washington DC, and also with the Design and Process Group, MSID, National Institute of Standards and Technology, Gaithersburg, MD 20899 USA (e-mail:
[email protected]). R. D. Sriram is with the Design and Process Group, MSID, National Institute of Standards and Technology, Gaithersburg, MD 20899 USA (e-mail:
[email protected]). K. W. Lyons is with the Nanomanufacturing Division of Design, Manufacture and Industrial Innovation National Science Foundation and also with the Design and Process Group, MSID, National Institute of Standards and Technology, Gaithersburg, MD 20899 (e-mail:
[email protected]). Digital Object Identifier 10.1109/TASE.2005.853583 1Virtual condition boundary: A constant boundary generated by the collective effects of a size feature’s specified MMC or LMC material condition and the geometric tolerance for that material condition.
I. INTRODUCTION OLERANCING a part, as is used in engineering practices, has two aspects: i) to assign tolerance values to a part so that the variations of its size/dimensions during manufacturing could be restrained/specified (tolerance synthesis), and ii) to assess how a part will behave in the final assembly (tolerance analysis) after it has been designed with some specified tolerance values. In general, there are four different types of tolerances: size, form, position and orientation. While size and form tolerances control the shapes of the features; orientation and positional tolerances control the relative orientation and location between features. In addition, three material conditions are defined as per standard ASME Y14.5 to these tolerances. They are maximum material condition (MMC), least material condition (LMC), and regardless of feature size (RFS). While MMC and LMC allow a bonus tolerance to the geometric tolerance specified when its feature size departs from its MMC/LMC, RFS does not. The modification to MMC or LMC results in a virtual condition boundary which controls the surface of the feature of size,2 while the modification to RFS results in a tolerance zone that controls the derived element,3 such as center plane, axis of a cylinder, center of a ball, etc. MMC is always used when a toleranced feature or a feature pattern mates with another feature or feature pattern on the other part. LMC is used when material preservation is of the main consideration. RFS is used when balance is important (for example, for parts that rotate). Whenever a tolerance specification requires a datum system to establish the positions or orientations of features with respect to other features, a datum reference frame (DRF) is provided. It may be a three-plane coordinate system defined by an ordered set of datum features. A datum feature is selected on the basis of its geometric relationship to the toleranced feature and the requirements of the design. To ensure proper assembly, corresponding interfacing features of mating parts should be selected as datum features. In addition, a datum feature should be accessible on the part and be of sufficient size to permit its use. The standardized methods for specifying tolerances use geometric symbols and numeric values. These symbolic specifications are convenient for interpretation by the designer. However,
T
2Feature of size: Feature of size is one cylindrical or spherical surface, or a set of opposed elements or opposed parallel surfaces, associated with a size dimension. 3Derived Element: Elements derived from a feature of size like: derived median line (from a cylindrical feature), derived median plane ( from a width-type feature), feature center point (from a spherical feature), feature axis (from a cylindrical feature), and feature center plane (from width-type feature).
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this form is not suitable for representation in the computer. One convenient way of representing the tolerance is to define a tolerance class with its attributes and methods as needed to cover an entire variety of tolerances [1]. While the tolerance class holds the tolerance information specified for each feature of a part, the need for converting these tolerance specifications to a generalized form arises from tolerance synthesis and analysis requirements. In order to take into account the effects of all components in an assembly in a global coordinate frame, a generalized coordinate space is needed so that each tolerance specification could be treated on the same footing. There are many approaches by different researchers to identify/use suitable coordinate systems. In the present case, the small displacement torsor (SDT) [2] has been adopted for representing the deviations of features from their nominal shapes/sizes. Other authors [3]–[6], [8] have also used different approaches using deviation parameters to model and represent geometric tolerances. From a geometric point of view, a tolerance allows some deviations of the nominal surface/feature, with reference to a datum. The variations could be intrinsic also. In both cases, it can be conceived as deviations of the nominal surface. Since these deviations are small compared to the dimensions of the part, such deviation of a surface/feature can be accurately described by two vectors: a three-component displacement vector and a three-component rotation vector defined at a convenient point on the feature [7]. Each vector is assumed to have a very small norm. These two components in a collective form are called an SDT. Assuming the two vectors as: small displacement and small rotation , the SDT . Even can be written as: though each SDT has six components, some of the components could be eliminated depending upon the type/nature of the feature the SDT represents. This is possible because most features used in engineering are regular. However, they may have some invariants along some of their six degrees of freedom. For example, for a cylindrical feature, there are two such parameters: axial rotation and axial movement; both keep the cylindrical feature invariant. Thus, these two parameters could be eliminated. This leads to a four-component deviation for a cylindrical fea. In ture with an axis along the -axis: general, two torsors are needed to completely define the position of any point on a feature; one for the displacement of the part in which the feature is defined–called a displacement torsor that could be conveniently defined at the center of a part; the other is to represent the intrinsic variation of the size of the feature—called an intrinsic torsor. For nonsize features,4 only the displacement torsor is required to represent the variation of the feature from its nominal position. Mapping a tolerance specification into the deviation space means expressing a given tolerance specification in terms of deviation parameters in the form of a set of inequalities. These inequalities are constraints that can be expressed as subspaces in the deviation space. This paper focuses on three types of fea4Nonsize feature: Nonsize feature is a surface having no unique or intrinsic size (diameter or width) dimension to measure, such as a nominally flat planar surface, a revolute–a surface, such as a cone, generated by revolving a spine about an axis, etc.
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tures: planar, cylindrical, and spherical. Since a planar feature is a nonsize feature, only size tolerance is considered. For cylindrical and spherical features, size, positional, and orientation are considered, as these are features of size. Positional tolerances are applied on an MMC, LMC, or RFS basis. In Section II, steps for mapping tolerance specifications to the deviation space are elaborated. In Section III, mapping procedures are described in detail for planar, cylindrical, and spherical features under various conditions. An example of the mapping procedure has been given in Section IV. II. STEPS FOR MAPPING TOLERANCE INTO DEVIATION SPACE The following steps are used to convert tolerance specifications of a feature into a set of inequalities in the deviation space. a) Generate the intrinsic torsor and the displacement torsor for every feature by eliminating the deviation parameters that are invariant for the feature to reduce the degrees of freedom. This reduces the number of deviation parameters needed to represent variation of the feature. For example, for a cylindrical feature, axial movement and axial rotation are two invariants and so the two parameters can be eliminated. b) Generate a virtual condition boundary (or a tolerance zone) based on the tolerance specification. In this step, the size of the VCB (or the tolerance zone) is computed for restricting the variation of the feature (or the derived) element, respectively. c) For a VCB, take an arbitrary point on the nominal surface of the feature and transform the point to a new position by applying the effect of the displacement torsors. on its For example, for a cylindrical feature, a point ; nominal surface is represented by two parameters , where and , . In case of tolerance zones, transform the whole derived element, such as the center plane, center axis, etc., to a new position by applying the effect of the displacement torsor. mentioned d) Eliminate the free parameters, [such as the in step c) above], by applying the condition that the extreme points of the transformed position should remain within the VCB or the derived element should remain within the tolerance zone. This step generates a set of inequalities connecting the deviation parameters with the tolerance specifications. The procedure for applying the above four steps has been elaborated in subsequent sections where the mappings for three basic features have been generated under various material conditions. The category of relations arising from the mapping of the deviation parameters of the features to corresponding tolerance values are generic nonlinear inequalities and takes the form: , where is some function of the devia, is some function of the tolerance tion parameters, is an index. Exact forms of parameters , and and depend on the nature of the feature/surface. For a planar surface, the above relationship becomes a set of linear constraints (planes in the deviation space). One interesting feature in the mapping relationships is the separable nature of the equations in terms of the variables and .
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III. MAPPING PROCEDURE The tolerances are specified on features or derived elements from feature of size. In this paper, mapping relations for one nonsize feature (planar feature) and two features of size (cylindrical feature and spherical feature) features are presented. Since it is assumed that the shape of a toleranced feature remains similar to that of its nominal feature (for example, a plane remains a plane, a cylinder remains a cylinder, etc.), the variations of position and orientation due to a form tolerance are very small [9]. In this work, only the size, orientation and positional tolerances have been considered. A geometric tolerance applied to a feature of size and modified to MMC, establishes a virtual condition boundary (VCB) outside of the material space adjacent to the feature. The feature should not cross this VCB. Likewise, a geometric tolerance applied to a feature of size and modified to LMC establishes a VCB inside the material and the feature should not cross the VCB. The size of the VCB is defined as the following [10]: Modified to MMC: For an internal feature of size: MMC virtual condition MMC size limit geometric tolerance. (1) For an external feature of size: MMC virtual condition MMC size limit geometric tolerance (2) Modified to LMC: For an internal feature of size: LMC virtual condition LMC size limit geometric tolerance. (3) For an external feature of size: LMC virtual condition LMC size limit geometric tolerance. (4) If a geometric tolerance applied to a feature of size is neither modified to MMC nor to LMC, it is modified to RFS by default (as per ASME Y14.5 standard). In this case, instead of VCB, the tolerance specification generates a tolerance zone in which the derived element should reside. A VCB or tolerance zone represents the basic intention of the designer that each point on the toleranced feature/derived element should remain within (or outside) this boundary/zone. Hence, the VCB or tolerance zone would be suitable for representing the relational limits on the deviation parameters associated with the deviation of the feature from its nominal position. Since the deviation parameters for a feature could be used to define the deviation of all points on a feature (by employing one or more independent parameters), the tolerance specifications could be thought of a set of limits for the deviation parameters and vice-versa.
Fig. 1.
Planar feature (rectangular) with size tolerance.
It has been established [8] that there are seven classes of symmetry groups for nominal features (continuous symmetry of geometric objects): spherical, cylindrical, plane, helical, revolute, prismatic and complex. In this paper, mapping relations for size, positional, and orientation tolerances for three features (planar, cylindrical, and spherical) have been established. While the mapping for the planar case is linear, those for the other two cases are nonlinear and the solutions in compact form for these cases could not be derived. Mapping for the remaining features (i.e., helical, revolute, etc.) are highly nonlinear and solution in compact form may not be obtainable. However, numerical solutions, similar to the cylindrical and spherical cases, should be possible. Symbols used in the following sections, unless otherwise specified, are defined as below: upper limit, size tolerance; lower limit, size tolerance; positional tolerance; perpendicularity tolerance; six components of the displacement torsor. A. Planar Features For each feature, a local coordinate system (LCS) is defined. The deviation parameters are defined in that LCS. For a planar surface, the LCS is: axis outward normal (emanating from the material side of the feature) and are local orthogonal coordinates on the plane. The equation for the nominal surface . For this plane, the deviation parame(plane) is given by . ters of SDT are Case 1: Rectangular ( ) planar surface. (Fig. 1) For the above feature, deviation parameters are , and the tolerance parameters are which are upper and lower tolerance limits, respectively, and the SDT is (5) Since this is a rectangular planar feature, the four cor, 0) have the ners with nominal coordinates ( , . Transmaximum deviation due to the effect of onto these four points by torsor transformation forming , four torsors on the four corner points are achieved as follows:
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Fig. 2.
Planar feature (circular) with size tolerance.
Comparing the -component with the specified tolerance value , the following four inequalities are obtained for the desired mapping:
Fig. 3. Cylindrical feature with LCS.
(6) (7) (8) (9) Case 2: Circular (radius ) planar surface. (Fig. 2) Deviation parameters Tolerance parameters For an arbitrary point on the circumference, at an angle with the -axis, , where is the pure positional variation of a point in the direction. The tolerance zone in this case is made up of and . should be within two planes: . This can be done by the above boundary for treating as a function of and then finding the extreme . This gives points by putting , which leads to the following mapping . This can (constraints) be written in parametric form as
Eliminating , gives
and
Fig. 4.
Cylindrical feature with tolerance modified to MMC.
where is the displacement torsor5 and is the intrinsic torsor.6 The VCB of the cylindrical surface is constructed by using the relations (1)–(4) and the VCB has the following properties: i) perfect shape as that of the nominal cylindrical feature; ii) size (diameter) of maximum size plus geometric tolerance, because its external feature and the positional tolerance is modified to MMC; iii) perfect orientation as that of the nominal cylindrical feature (vertical to datum ); iv) perfect location as that of the nominal cylindrical feature relative to datum and . , , ) (Fig. 3) is any point on the nominal Point ( cylindrical surface. After transformation due to the effect of the two torsors, the new position of the transformed point can be calculated as below
B. Cylindrical Feature–Modified to MMC A cylindrical feature with a local coordinate system is shown in Fig. 3 and the tolerance specification is shown in Fig. 4. For this surface, the deviation parameters of the SDT are and , and the tolerance param. Based on this notation, constraints eters are can be derived connecting these parameters with the specified tolerance values as detailed below 5Displacement torsor: torsor for the deviation of the feature from the nominal position. 6Intrinsic torsor: torsor to represent the intrinsic variation of size of the feature.
WANG et al.: SCHEME FOR MAPPING TOLERANCE SPECIFICATIONS TO GENERALIZED DEVIATION SPACE
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The 4 4 matrix on the left-hand side (LHS) of the equation is the transformation matrix for the point following the asby sumed transformation sequence the displacement torsor. The 4 by 1 matrix (vector) on the LHS of the equation represents the changed position of the point by the intrinsic torsor. So the new position (Fig. 3) is given by
Hence, the constraint equation becomes
(10) Fig. 5.
Cylindrical feature with tolerance modified to RFS.
and . The LHS is a function of where two independent parameters and . The inequality is valid for and . and are eliminated from the LHS of (10) by solving the maximum of the LHS function. LHS of (10) is then written as:
In the case of the cylinder, because of the rigid motion, points that move the most are those on the two ending circle. Hence, and . we only need to consider two cases: The above maximization problem of two variable functions is reduced to a maximization problem of one variable function Eliminating , we get
Fig. 6. Cylindrical tolerance zone.
C. Cylindrical Feature Modified to RFS
The final set of constraints are
A cylindrical feature modified to RFS is shown in Fig. 5. Fig. 6 shows a cylindrical tolerance zone (to control the cylindrical feature’s center axis), which may be due to positional, concentricity and/or run-out tolerances for any cylindrical surface. A local coordinate system (LCS) is defined as shown in Fig. 6. The displacement torsor and the intrinsic torsor are
When the toleranced feature is internal as, for example, for a hole, the constraints are
If the external feature is modified to LMC, the constraints are
If the internal feature is modified to LMC, the constraints are
The difference between this case and the one in Fig. 4 is that is independent from . It means that , the radius of tolerance zone, does not change due to the variation of , which is the departure from the nominal radius. Because it is the axis of the feature of size that is toleranced but not the surface of the feature of size, a different method is used to generate the constraints on mapping deviation parameters to the tolerance and are independent, these two need to be zone. Since treated separately. For , there are four deviation parameters and . The axis of the cylinder can be conas shown in Fig. 6. Under rigid sidered as a line segment
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The 4 4 matrix on the LHS of the above equation is the transformation matrix for any point on the spherical surface due to the displacement torsor. The 4 by 1 matrix (vector) on the LHS of the equation represents the changed position of the point due to the intrinsic torsor. The constraint equations are Fig. 7.
Fig. 8.
Spherical feature with LCS.
Spherical feature with tolerance specification.
motion, (0,0, ) or (0, 0, the transformed point of (0,0, ) (
The constraint on
where
or
) move the most. Hence, or ) is
and
is
, and
D. Spherical Surface For a spherical surface, the LCS is a axis along radius of the sphere and are local orthogonal coordinates at the center of the sphere (Figs. 7 and 8). For this surface, the deviation paand . Based on rameters of the SDT are this notation, constraints are derived connecting these parameters with the specified tolerance values as detailed as follows. The VCB of the spherical surface is constructed with the following properties: i) perfect shape as that of the nominal spherical feature; , because it is an external ii) size (diameter) of feature and the positional tolerance is modified to MMC; iii) perfect location as that of the nominal cylindrical feature relative to datum and . Applying the same procedures as they have been followed in the case of cylindrical feature modified to MMC, we get the equation shown at the bottom of the page.
The above inequality is valid for and . This is a problem to find the maximum of the and ). above function in the area ( is the location of a maximum of A point only if is a critical point of this function. Similarly, a is the location of a minimum of only point is a critical point of this function. if A critical point of on is one of • boundary point of ; • point in where all of the first partial derivatives of are zero. A critical point of the second kind is either a local maximum, or a local minimum, or a saddle point. The second partial test, Hessian determinant , can tell what it is. has continuous second partial derivaSuppose that and that tives near
Let . Then • if and , then is a local maximum; • if and , then is a local minimum; , then has a saddle point at ; • if , any one of the three is still possible. • if Elimination of parameters and are done by maximizing the following function:
where
WANG et al.: SCHEME FOR MAPPING TOLERANCE SPECIFICATIONS TO GENERALIZED DEVIATION SPACE
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Solving the above two equations in symbolic form (using Matlab 6) results in four sets of solutions. The respective Hessian determinants are
The respective
values are
Fig. 9. Effect of
1 ( ) on (8 9). z d
f
;
The final set of constraints is
When a toleranced feature is internal, a spherical hole for example, the final constraints are
where , , , , and “ ” mean power. Hence, we know that the last two solutions are saddle points because their Hessian determinants are always negative. totally depends on the sign of “ ” which is . The sign of “ ” is always positive as is much larger than . All other items is positive, the first two solutions are positive. Therefore, if and in the following] are local max[termed imum. If is negative, the first two solutions are local minimum. This can be illustrated by substituting values to and plot in Matlab (Fig. 9). and ], we can find In area [ and depending on the sign of . Points on the boundary are also critical points as described before. The boundary con, , , and . sists of four planes: , we get four one-variable Substituting these values into functions
If the external feature is modified to LMC, the constraints are
If the internal feature is modified to LMC, the constraints are
For a spherical feature modified in RFS, the and variation , , and , deviations of size of feature are independent of of the position of the feature. Only center “O”(0,0,0) (Fig. 7) of the sphere is controlled in a spherical tolerance zone with a . The transformed point of “O” is radius of
Therefore, the constraints are and
E. Special Cases of Tolerance Specifications Eliminating
In summary
from the last two functions
1) Composite Tolerance Specification: Sometimes more than one tolerance is specified on one feature of size. In Fig. 10, the cylindrical feature is controlled by both positional and perpendicular tolerances. In those cases, more constraints need to be generated. The ASME Y14.5 standard requires that all of the tolerances specified to one feature of size be satisfied independently. This means that every tolerance needs to be treated separately. For the above example, the constraints for a positional tolerance are same as those in Section III-B (cylindrical feature–modified to
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From Section III-B, for a positional tolerance of similar specand , which are the maximum and minimum ification, , are computed. of function The final constraints due to the specified perpendicular and positional tolerances are
Fig. 10.
When the toleranced feature is internal (e.g., for a hole), the constraints are
Composite tolerance specification.
MMC) of Section III. The following is the derivation of the constraints for the perpendicular tolerance. The role of the perpendicular tolerance here is to give more constraints on the orientation of the feature. Because perpendicular tolerance does not fix the position of the virtual condition boundary and the tolerance only controls orientation, the position of the transformed point due to the perpendicular tolerance can be calculated as follows:
If the external feature is modified to LMC, the constraints are
If the internal feature is modified to LMC, the constraints are So the constraint equation from the specified perpendicular tolerance is
(11) The constraint equation from the specified positional tolerance is
(12) Equations (11) and (12) are the constraints due to the composite tolerance specification in the above example. Using the same procedures as in Section III-B (cylindrical feature–modified to mmc) to eliminate and , the following constraints are obtained on the deviations for the perpendicular tolerance: where By equating the partial derivatives of and and and
to zero
we get
, where is maximum of function is minimum of the function .
2) MMC Specification on Datums: Sometimes MMC is specified on datum feature (Fig. 11), which is itself a feature of size. It allows the feature of size, which is toleranced to the datum feature, to have more variations when the datum feature departs from its MMC. This specification does not increase the size of the VCB of the toleranced feature of size but allows the VCB’s location to move if the datum feature of size departs from its MMC. The amount of the allowed movement is equal to the difference of the datum feature from its MMC. In contrast to the composite tolerance specification as shown in Fig. 10, the position of the VCB of the LHS cylindrical feature is not fixed. The allowed movement of its VCB is dependent on the departure of the size of the right-hand side (RHS) cylindrical feature from its MMC. This problem can be modeled as if the positional tolerance and perpendicular tolerance are specified on the left-hand fea7 and ture. The positional tolerance will be . Following the procedures the perpendicular tolerance is as described in Section III-B, the constraints on deviations of 7In
TU + Tp 0 dr) is from the other cylindrical feature.
this case, (
WANG et al.: SCHEME FOR MAPPING TOLERANCE SPECIFICATIONS TO GENERALIZED DEVIATION SPACE
Fig. 11.
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MMC specified to a datum.
Fig. 13.
Ring gear with tolerance specifications [12].
. The 4 8 - 32 thread hole pattern is controlled by a positional tolerance with datum and (modified to MMC) as its DRF. . The size tolerance of the The positional tolerance value is . four thread holes is The radius of the thread hole is given as 0.125/2 and that of the pinhole is 0.315/2. The deviations of the radii are assumed and for the radius of the thread hole and the radius to be of the pinhole, respectively. Constraints on deviation parameters of the two pinholes by the perpendicular tolerance and the size tolerance are Fig. 12.
Exploded view of a gearbox assembly.
the LHS cylindrical feature could be generated by substituting and
IV. EXAMPLE Fig. 12 shows a gearbox assembly. In order to show how the tolerance mapping works, one component, the ring gear, is chosen from the assembly. The constraints on deviation parameters are generated from tolerance specifications as shown in Fig. 13. 0.125 pinhole pattern is controlled by In Fig. 13, the 2 a perpendicular tolerance with the datum . The datum is constructed by a planar surface on the ring gear, which mates with the output housing. The perpendicular tolerance value is modified to MMC. The size tolerance of the two pinholes is . The 2 0.125 pinhole pattern defines datum
(13) is a function of nominal dimensions and deviawhere tion parameters. Constraints on deviation parameters of the four thread holes by the positional tolerance and the size tolerance are
(14) where and are functions of nominal dimensions and deviation parameters. The constraints in (13) and (14) can be used for tolerance synthesis and analysis. Tolerance synthesis is normally modeled as an optimization process to minimize the production cost subject to functional constraints and assemblability constraints [11]. Deviation parameters on each toleranced feature are returned as optimization , , and can be achieved by results. Values of substituting the returned deviation parameters.
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From equation (16), three inequalities are achieved
The above inequalities construct four half spaces in space. The zone generated by the four , , and . half spaces represents solution space for The solution space could be restricted by more constraints on the tolerance values, such as process capability of available processes on the tolerance values. , , and can be achieved from (14). Solutions of In a tolerance analysis problem, engineers are interested in 1) the propagation of geometric variations of toleranced features (stack up analysis in the assembly); and 2) what the effects of those variations are on functional requirements. Because tolerance values are known, constraints in (13) and (14) provide a mathematical relationship between tolerance specification and deviation parameters. The constraints derived from (13) and (14) can be used in tolerance analysis as well. V. CONCLUSION The mathematical representation of geometric tolerances is very important for computerized tolerance synthesis and analysis. Both tolerance synthesis and analysis need the mapping between deviation parameters of the toleranced features and the tolerance specification in standard formats. While tolerance synthesis needs the mapping so that the deviation parameters can be mapped to the tolerance value solutions, tolerance analysis needs the mapping so that effects of the tolerance specification on deviation parameters can be represented. In this paper, a method has been presented for transforming tolerance specifications given by ASME Y14.5 into constraints in a generalized coordinate frame (deviation space). The mapping relations for three features (plane, cylinder, and sphere) under various tolerance specifications have been generated. These mapping relations have been used for tolerance synthesis work [11] and further study would be needed for using these relations in tolerance analysis. ACKNOWLEDGMENT No approval or endorsement of any commercial product by the National Institute of Standards and Technology is implied. Certain commercial equipments, instruments, or materials are identified in this work in order to facilitate a better understanding. Such an identification does not imply endorsement or recommendation by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose. REFERENCES [1] U. Roy, N. Pramanik, R. Sudarsan, R. D. Sriram, and K. W. Lyons, “Function-to-form mapping: Model, representation and applications in design synthesis,” J. Comput.-Aided Design, vol. 33, no. 10, pp. 699–720, Sep. 2001.
[2] E. Ballot and P. Bourdet, “A computation method for the consequences of geometric errors in mechanisms,” in Geometric Design Tolerancing: Theories, Standards and Applications, H. A. Elmaraghy, Ed. London, U.K.: Chapman & Hall, 1998, pp. 197–207. [3] E. Ballot, P. Bourdet, and F. Thiebaut, “3D tolerance chains: Present conclusions and further research, integration of functionality specification and verification,” in Proc. 7th CIRP Int. Sem. Computer Aided Tolerancing, ENS de Cachan, France, Apr. 24–25, 2001. [4] S. Samper and M. Giordano, “Simultaneous method for tolerancing flexible mechanisms,” in Proc. 7th CIRP Int. Sem. Computer Aided Tolerancing, ENS de Cachan, France, Apr. 24–25, 2001. [5] T. Denis, D. Vincent, and C. Yves, “Operations on polytopes: Application to tolerance analysis,” in Proc. Global Consistency of Tolerances, 6th CIRP Int. Sem. Computer Aided Tolerancing, Enschede, The Netherlands, Mar. 1999, pp. 425–434. [6] J. K. Davidson, A. Mujezinovic, and J. J. Shah, “A new mathematical model for geometric tolerances as applied to round faces,” in Proc. ASME Design Engineering Techn. Conf., Baltimore, MD, Sep. 2000, pp. 963–979. [7] M. Bourdet et al., “The concept of small displacement torsors in metrology,” in Advanced Mathematical Tools in Metrology II: World Scientific, 1996, vol. 40, Series on advances in mathematics for applied science. [8] V. Srinivason, “A geometrical product specification language based on a classification of symmetry groups,” J. Comput. Aided Design, vol. 31, pp. 659–668, 1999. [9] O. L. Gilbert, “Representation of geometric variations using matrix transforms for statistical tolerance analysis in assemblies,” M.S. thesis, Mass. Inst. Technol., Cambridge, MA, 1992. [10] P. Drake, Dimensioning and Tolerancing Handbook. New York: McGraw-Hill, 1999. [11] N. Pramanik, U. Roy, R. Sudarsan, R. D. Sriram, and K. W. Lyons, “Synthesis of geometric tolerances of a gearbox using a deviation-based toleance synthesis scheme,” in Proc. ASME Design Engineering Technical Conf. Computers Information Engineering Conf., Chicago, IL, Sep. 2–6, 2003, DETC2003-CIE-48 192. [12] H. Wang, N. Pramanik, U. Roy, R. Sudarsan, R. D. Sriram, and K. W. A. Lyons, “Functional tolerancing of a gearbox,” in Proc. North American Manufactuing Research Conf., Hamilton, ON, Canada, May 20–23, 2003.
Haoyu Wang received the B.S. degree in mechanical engineering from Hebei Institute of Technology, Tianjin, China, in 1994, the M.S. degree in mechanical and electrical engineering from Harbin Institute of Technology, Harbin, China, in 1996, and the Ph.D. degree (Hons.) in mechanical engineering from Syracuse University, Syracuse, NY, in 2004. Currently, he is an Assistant Professor in the Department of Applied Sciences, Firelands College of Bowling Green State University, Huron, OH, where he has been since 2004. His main research interests are in computer-aided design/computer-aided manufacturing, tolerance analysis, and synthesis, quality control, three-dimensional motion analysis, and injury biomechanics.
Nilmani Pramanik received the B.S. degree in mechanical engineering from Jadavpur University, Kolkata, India; the M.S. degree from Birla Institute of Technology and Science (BITS), Pilani, India; and the Ph.D. degree in mechanical engineering from Syracuse University, Syracuse, NY. Currently, he is an Assistant Professor of Manufacturing Technology in the Department of Industrial Technology, University of Northern Iowa, Cedar Falls. He has more than 15 years of industrial experience in engineering design, analysis, and development of engineering and Enterprise Resource Planning (ERP) software. His areas of interest are computer-aided design/computer-aided manufacturing systems, geometric tolerancing, mathematical modeling, object-oriented programming, expert systems, and AI applications.
WANG et al.: SCHEME FOR MAPPING TOLERANCE SPECIFICATIONS TO GENERALIZED DEVIATION SPACE
Utpal Roy is a Professor with the Mechanical, Aerospace, and Manufacturing Engineering Department, Syracuse University, Syracuse, NY. His current research interests include geometric tolerance analysis and synthesis, knowledge-based computer-aided engineering, and product development.
Rachuri Sudarsan received the M.S. and Ph.D. degrees from the Indian Institute of Science, Bangalore, in 1989 and 1992, respectively. Currently, he is a Research Professor with the Department of Engineering Management, George Washington University, Washington DC. He is a Guest Researcher in the Design and Process Group, Manufacturing Systems Integration Division, National Institute of Science and Technology (NIST), Gaithersburg, MD. His work at NIST includes the development of information models for product life-cycle management, assembly models and system level tolerancing, and standards development. He coordinates research projects with industry and academia and works closely with various standard bodies including ISO TC 184/SC4. Dr. Sudarsan is a member of ASME Y14.5.1 and and is an advisory group member for ISO/TC 213/AG12, Mathematical Support for global positioning systems. He is the Regional Editor (North America) for the International Journal of Product Development, and Associate Editor for the International Journal of Product Lifecycle Management. His areas of interest include scientific computing, mathematical modeling, product life-cycle management, ontology modeling, system-level tolerancing, quality, object-oriented modeling, and knowledge engineering.
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Ram D. Sriram (SM’00) received the B.S. degree from the Indian Institute of Technology, Madras, India, in 1980, and the M.S. and Ph.D. degrees from Carnegie Mellon University, Pittsburgh, PA, in 1981 and 1986, respectively. He currently leads the Design and Process group in the Manufacturing Systems Integration Division, National Institute of Standards and Technology, where he conducts research on standards for interoperability of computer-aided design systems and on health-care informatics. Previoulsy, he was on the engineering faculty of the Massachusetts Institute of Technology (MIT), Cambridge, from 1986 to 1994 and was instrumental in setting up the Intelligent Engineering Systems Laboratory. He also initiated the MIT-DICE project, which was one of the pioneering projects in collaborative engineering at MIT. He has co-authored or authored more than 175 papers, books, and reports in computer-aided engineering, including 13 books. He was a founding co-editor of the International Journal for AI in Engineering. In 1989, he was the recipient of a Presidential Young Investigators Award from the National Science Foundation U.S.
Kevin W. Lyons received the B.S.M.E. degree from the University of Florida, Gainesville, and the M.S.M.E. degree from the University of Maryland, College Park. Currently, he is Program Director with the Nanomanufacturing Program at the National Science Foundation (NSF), Arlington, VA. The program emphasizes the scale-up of nanotechnology for high rate production, reliability, robustness, yield, efficiency, and cost issues for manufacturing products and services. He oversees two existing Nanomanufacturing Centers and a third center that will be selected in Fall 2005. Prior to his NSF assignment, he was a Program Manager with the Manufacturing Engineering Laboratory at the National Institute of Standards and Technology, Gaithersburg, MD. His responsibilities included research in nanomanufacturing, assembly, virtual assembly, and rapid prototyping. From 1996 to 1999, he was Program Manager with the Defense Advanced Research Projects Agency (DARPA), where he was responsible for the conceptualization, development, and execution of advanced research-and-development programs in design and manufacturing. From 1977 to 1992, he worked in the industry in various staff and supervisory positions dealing in engineering marketing, product design and analysis, factory automation, and quality engineering.
