Tolerance Analysis of Machining Fixture Locators

S. A. Choudhuri Department of Industrial and Manufacturing Engineering E. C. De Meter Department of Industrial and Manufacturing Engineering, Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA 16802 Tolerance Analysis of Machining Fixture Locators The geometric variability of locators within a machining fixture is a known source of datum establishment error and machined feature geometric error. A locator tolerance is used to specify the range of permissible locator variation. Currently there are no models that relate a locator tolerance scheme to the worst case geometric errors that may result due to datum establishment error. This paper presents a methodology for modeling and analyzing the impact of a locator tolerance scheme on the potential datum related, geometric errors of linear, machined features. This paper also provides a simulation study in which locator tolerance analysis is applied to reveal some important insights into the relationship between machined feature geometric error, locator design, and locator tolerance scheme. 1 Introduction Machining fixture locators are used to establish the datum reference frame (x-y-z) of a workpiece with respect to the axial reference frame {X-Y-Z) of a machine tool as illustrated in Fig. 1. Once a workpiece has been located and clamped, the cutting tool is moved relative to X-Y-Z for the purpose of generating the machined surfaces of the workpiece. Assuming that the workpiece datum features and the locator surfaces are perfect and that no other sources of machining error exist, the machined surfaces will be true with respect to form and position in relation to the workpiece datum reference frame. Alternatively, if the geometries of the locators are allowed to vary from their true form, there will be a misalignment between x-y-z and X -Y-Z as shown in Fig. 2. This misalignment is termed datum establishment error. In general datum establishment error will not result in form errors of the machined surfaces. However it will result in datum related errors such as orientation errors, position errors, and profile errors. The manner by which the geometry of a locator may vary is controlled by a tolerance as shown in Fig. 3. This tolerance appears in the design specifications of the fixture and is assigned by the fixture designer. Different tolerance schemes can be employed to control the geometry of a locator. For example, the contact surfaces of two of the locators in Fig. 3 are controlled by a dimensional tolerance. Alternatively the third locator is controlled by a profile of surface tolerance. Of practical interest to the fixture designer is the relationship between the datum establishment errors permitted by a locator tolerance scheme and the worst case geometric errors that may result. In recent years, a number of datum establishment error models have been developed. Bourdet and Clement (1974) proposed the use of a displacement screw vector to mathematically describe datum establishment error. In addition they developed a model to relate a screw displacement vector to known deviations of the positions of the contact points between a prismatic workpiece and a system of spherical tip locators. This model relies on the assumption that a deviated contact point lies on the nominal axis of the locator. Bourdet and Clement (1988) extended this work by developing a model to determine the nominal positions of locators that will minimize the magnitude of the screw displacement vector. Laloum et al. (1989) and Weil et al. (1991) subsequently investigated a variety of optimization techniques that could be employed to solve Bourdet and Clement's model. In Contributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received Dec. 1996; revised March 1998. Associate Technical Editor: T. C. Woo. addition Weil et al. (1991) suggested that a function could be developed to relate the screw displacement vector to the geometric variation of a critical feature, and that this function should be minimized instead of the magnitude of the screw displacement vector. The problem that currently exists is that the models developed thus far ignore the contact geometry between the locators and workpiece. Consequently they can not be used to investigate the impact of a locator tolerance scheme on datum establishment error. Likewise they do not provide a methodology of predicting the impact of a known datum establishment error on the potential geometric error of a machined surface. The objective of this paper is to present a general methodology for modeling and analyzing the impact of a locator tolerance scheme on the potential geometric errors of linear, machined features. Examples of such features include planar surfaces and hole axes. The scope of this paper is limited to dimensional and profile tolerances applied to spherical tip locators, planar workpiece datum features, and linear, machined features that are bounded by planar workpiece surfaces. In addition, it is assumed that all other potential sources of workpiece geometric errors are ignored, including datum feature variability. In the sections that follow, this paper presents a model that relates datum establishment error to locator geometric variability. It subsequently describes the tolerance zones associated with a profile tolerance scheme and dimensional tolerance scheme, and illustrates how the parameters associated with these zones can serve as input to the datum establishment error model. The paper subsequently presents a methodology that can be employed to predict the impact of a known datum establishment error on the geometric errors of linear, machined features. It further illustrates how this model can then be used to link a locator tolerance scheme to the worst case geometric errors that may result from datum establishment error. The paper then provides a simulation study in which the locator tolerance analysis models are applied to a fixture-workpiece system. As will be discussed, the results of this case study provides valuable insight into the interactions between locator design variables, locator tolerance schemes, and machined feature geometric error. 2 Locator Tolerance-Datum Establishment Analysis A locator tolerance defines the extremes by which the geometry of a locator may vary. By geometry it is meant that critical portion of the locator in potential contact with the workpiece. To predict the impact of a locator tolerance on datum establishment error, two models are presented. The first model relates locator Journal of Manufacturing Science and Engineering MAY 1999, Vol. 121 / 273 Copyright 1999 by ASME

machined surfaces- r~\ 0.02 C D r3x SR6.25 I Bi Y y x *T H workpiece! 25.0 ±0.01 machining fixture Fig. 3 Locator tolerance schemes Fig. 1 locators Ideal datum establishment error to datum establishment error. The second model relates a locator tolerance scheme to extreme locator error and its resultant impact on datum establishment error. The locator errordatum establishment error model is described next. 2.1 Locator -Datum Establishment Model. Consider the case in which no locator error exists as illustrated for the ith locator in Fig. 4. Here the geometry of the locator is a perfect hemisphere with nominal center, c oi, and nominal radius, rad,. Note that the workpiece datum reference frame, x-y-z, and the machine tool axial reference frame, X-Y-Z, are coincident. In addition the point of contact between the workpiece and the ith spherical tip locator is also shown. The coordinates of this point are defined by the vector, p,. Likewise the coordinates of the unit surface normal at this point are defined by the vector n 0,. Now assume that the geometry of the locator deviates to an extreme allowed by a tolerance as illustrated in Fig. 5. As will be discussed shortly, the geometry of the locator will remain spherical but will now have the center point coordinates, c,, and radius, rad,. Due to this variation, datum establishment error will result. This error can be defined by the translation vector, d, and the rotational vector, 9. In addition, the point of contact between the workpiece and the ith locator will also change. The coordinates of this point are defined by p,. Likewise the coordinates of the surface normal at this point are defined by the vector, n,. Please note that n* is not necessarily a unit normal. Assuming that the original point of contact is fixed to the workpiece, the coordinates of this point and the original surface normal are now defined by p di and n di, respectively. In general the new point of contact must lie on the surface of the locator. Consequently p, must satisfy the following equation: rad? = 0 (1) Likewise the new point of contact must also lie on the planar surface of the workpiece. Consequently p, must also satisfy: (P.- - Vdi) n dl - = 0 (2) Finally the new point of contact should be a point of tangency between the workpiece surface and the locator. This tartgency relationship is defined by the following equation: n di = a,-n,-, (3) where a, is an arbitrary scalar quantity. As was mentioned previously, the original point of contact between the workpiece and the ith locator is assumed to be fixed to the workpiece. Consequently p di and n di are related to d and 6 by the following coordinate transformation equations: p di = Rp, + d (4) n di = R n oi (5) where R is the rotation matrix that defines the orientation of x- y-z relative to X-Y-Z. R is a function of 9. In general 9 is nominal machined surfaces profile error nominal locator surfaces locators Fig. 2 Datum establishment (D. E.) error 274 / Vol. 121, MAY 1999 Transactions of the ASME

/ / upper tolerance zone nominal locator surface z rad or^ ' n oi lower tolerance zone Fig. 4 z K * L Poi / tf X c oi Workpiece-locator contact without D. E. error usually very small in practice. Consequently it is assumed that the rotation matrix can be linearized as follows: R = 1 -t (6) Let/(p,, c,, rad,) be a function that represents the left hand side of Eq. (1). The normal to the locator surface at the new point of contact can be evaluated by computing the gradient f /(P<> c ;> rad,) at that point. This results in the following constraint: n, = V/(p,-, c,, rad,) (7) Through algebraic manipulation, Eqs. (1-7) can be combined for each locator. This will lead to the following contact constraints for the fixture-workpiece system: (pf - c x ) 2 + (pj - c y ) 2 + (pf - cf) 2 - rad? = 0 (n z oi + n y oi * 9 X - n' oi * 6 y )*(d z + p y oi * 0* -p x oi * 9 y + P z a-p z l) + (n y oi - n z oi * 0* + n x oi * 9 Z ) *(d y -p oi *9 x + p oi *9 z +p y i-p y ) + (n x oi + n z oi*9 y -n y oi*9 z ) *{d x +p z oi*9 y - p y oi *9 z +p x oi- pi) = 0 (n x oi + n z oi * e y - n y oi * 9 Z ) - 2a,- * (p x -c x ) = 0 (n y oi - n z oi *9 X + n x, * 9 Z ) - 2a-, * (p y -c y ) = 0 (n z oi + n y oi 6 x - n' oi * 6 y ) -2a,* (p z -c z ) = 0 for / = 1, 6 (8) The equations defined by [8] are nonlinear and implicit in form. However their relationship to the fixture-workpiece system should be interpreted as follows. The constants in [8] are p i and n oi. These values define the contact geometry between the locators and workpiece when the locators are at true form (i.e. no geometric error). The independent variables in [8] are Fig. 6 Locator profile tolerance zone c, and rad,. These values define the actual geometry of the locators. The dependent variables are d, 9, p,, n,, and a,. These values define the datum establishment error and contact geometry that will result if c, and rad, deviate from their true (nominal) values. In general [8] is comprised of thirty equations. Likewise if c { and radj are set to known values, [8] will consist of thirty unknowns (d, 9, p,, n,, and a,). The equations in (8) can be solved for the unknowns using a numeric, nonlinear equation solution technique. In general if c ; and rad, are set to c oi and rad, respectively, the solution of [8] will yield: d = 9 = [0] p, = p, for i = 1.. 6 n, = n 0, for i = 1.. 6 (9) (10) (11) Alternatively if c, and rad, are allowed to deviate from their nominal values, the solution of [8] will lead to nonzero d and 9, and deviations of p, and n, from p, and n 0, respectively. The manner and degree by which c ; and rad, are allowed to vary from their nominal values is controlled by the tolerances applied to the locators. This is discussed next. 2.2 Locator Tolerance Datum Establishment Model. A locator tolerance defines a three dimensional tolerance zone within which the critical geometry of a locator must lie. The form of the tolerance zone is dependent upon the type of tolerance employed. The size, 7", of the tolerance zone is specified in the call out. For example consider the profile tolerance zone (as defined by ANSI 1995) illustrated in Fig. 6. The tolerance zone consists of two hemispheres. Both are centered about c oi. The radius of the upper zone, rad,, and the radius of the lower zone, rad,, are defined as: rad u/ = rad, + Til. rad/, = rad, 772. (12) (13) Alternatively consider the dimensional tolerance zone illustrated in Fig. 7. As before, the tolerance zone consists of two hemispheres. However in this case, both have the same radius, upper tolerance zone nominal locator surface rad oi lower tolerance zone Fig. 5 Workpiece-locator contact with 0. E. error Fig. 7 Locator dimensional tolerance zone Journal of Manufacturing Science and Engineering MAY 1999, Vol. 121 / 275

rad 0,. However each is centered about a different point. The center of the upper zone, c,, and the center of the lower zone, c u, are defined as: c ul, <= c oi + TI2*n oi (14) c : <= c oi - T/2*n oi (15) At its extreme, the geometry of a locator will take the form of either the upper or lower tolerance zone. It is assumed that the datum related errors of linear, machined features take on their greatest values when the geometry of each locator is allowed to vary to its extreme. Consequently it is necessary to evaluate datum establishment error at all possible combinations of locator extremes. Since a locator may take on the form of either the lower or upper tolerance zone, and considering the fact that a fixture utilizes six locators, the number of extreme cases to be evaluated is 2 6 = 64. The validity of this assumption is indicated by the results of the simulation study presented in section 5. Here it is seen that despite the non linearity of the governing equations, the sizes of the datum related geometric errors of the planar surfaces vary linearly with the sizes of the tolerances. For a profile tolerance scheme, determination of the datum establishment error for each case requires that each rad, be set to the appropriate value of rad u/ or rad,,. Subsequently the nonlinear equations defined in [8] are solved for d and 9. It is recommended that the no variability solution, as defined by Eqs. (9-11) be used as the starting point for the solution procedure. The starting values of a t should be set to unity. The approach used for a dimensional tolerance scheme is the same with the exception that c, are set to either c u, or c«, while rad; are set to their nominal values. Having determined the datum establishment error for each case, the next step in the process is to determine the impact of the tolerance scheme on the resultant geometric errors of the machined surfaces. This is discussed in the following section. However prior to this discussion, one additional attribute of a tolerance scheme will be discussed. This is tolerance zone volume. In general the larger the volume, the more room locator geometry has to vary. For example, the volume, V p, occupied by a profile tolerance zone can be defined as: V p = 27r(rad^, T + T 3 /12) ~ 2TT rad?, T (16) In turn the volume, V d, occupied by a dimensional tolerance zone can be computed as: V d = 7rrad*,r (17) Consequently for a given tolerance size, a profile tolerance will provide twice as much room for variation as a dimensional tolerance. The importance of this will be addressed in section 5. Fig. 8 Surface generation without D. E. error to perform this procedure for any linear, machined feature will be provided. 3.1 Step 1: Machined Feature Characterization. Consider the example shown in Fig. 8. The machined, planar surface is generated without the presence of datum establishment error. The machined surface lies on the cutting plane swept by the face mill. In addition, the machined surface lies within the boundaries defined by the intersection of the cutting plane and the sides of the workpiece. Since the workpiece is prismatic, the resultant machined surface is convex. As such, knowledge of the position vectors, s,-, of the extreme points of this surface is sufficient to characterize it. This is due to the fact that the position vector, s, of any point on the surface can be represented as a convex combination of s,. Let K represent the number of extreme points of the machined surface. The representation of s can be expressed as: K K s = L M,; lx, = l; 0<\,<1 (18) where X, is an arbitrary scalar. In general, each extreme point will lie at the intersection of the cutting plane and two sides of the workpiece. Now consider the case in which the machined surface is generated in the presence of datum establishment error as shown in Fig. 9. The resultant machined surface is still convex. However its orientation and position with respect to the workpiece datum reference frame has changed. In addition the extreme points of the surface have changed as well. The position vectors of these new extreme points are defined as s,. As in the case of no datum establishment error, the position vectors of these new extreme points can be computed as the intersection of the cutting plane and the sides of the workpiece. 3 Locator Tolerance-Workpiece Geometric Analysis To predict the impact of a locator tolerance scheme on the geometric error of a linear, machined feature, a three step procedure is employed. The first step in the procedure is to characterize the geometry of the machined feature with the assumption that it is generated in the presence of a known datum establishment error. The second step is to assess the geometric accuracy of the feature. Since a given tolerance scheme will result in sixty-four distinct cases of datum establishment error, the third step is to determine the case at which maximum geometric error occurs. For the sake of simplicity, this procedure will be illustrated for planar surfaces only. However the general steps necessary 276 / Vol. 121, MAY 1999 Fig. 9 Surface generation with D. E. error Transactions of the ASME

upper minimum zone nominal surface lower minimum zone Fig. 10 error minimum zone However in the case of datum establishment error, the position and orientation of the workpiece sides have been altered as well. Consequently these surfaces must be characterized prior to computation of the extreme points of the machined surface. This may be done by noting that the equation of the plane in which the side of a workpiece resides can be derived from three arbitrary, non-collinear points on the side. For example in Fig. 8, the equation of the plane in which the front side of the workpiece resides can be derived from the position vectors q!, q 2, and q 3. In the advent of datum establishment error, these points undergo a rigid body displacement from their nominal positions as illustrated in Fig. 9. The position vectors of these displaced points are r,, r 2, and r 3. These vectors can be computed using the following relationship: r, = R q, + d (19) Having performed this procedure and the subsequent computation of s,, one final task must be performed, s, are the position vectors of the machined surface relative to the machine tool axial reference frame, X-Y-Z. However the tolerances applied to the machined surface are defined relative to the workpiece Fig. 12 Fixture-workpiece system datum reference frame, x-y-z. Consequently the position vectors, u,, of the extreme points relative to this frame must be computed. This is accomplished by applying the following coordinate transform: R 's, R (20) In order to make it applicable for the geometry's of other linear, machined features, such as hole axes, that are bounded by planar workpiece surfaces, the procedure just described can be generalized as follows: (1) identify the general mathematical form of the swept cutter path that defines the unbounded form of the linear, machined feature (2) identify q 1; q 2, and q 3 for each workpiece side that intersects the swept cutter path (3) for known d and 8, compute r,, r 2, and r 3 for each workpiece side and derive the corresponding plane equation (4) mathematically characterize the boundaries between the /_? A r\? A B c Machined Surface 1 /_? A C\? A B C Machined Surface 2 Datum C BA ^yj,, Datum A 0 Datum B Fig. 11 Finished workpiece Journal of Manufacturing Science and Engineering MAY 1999, Vol. 121 / 277

locators respect to datum A is similar. Like the previous example, the upper and lower zones are planes mutually parallel to the datum A plane. However unlike the profile minimum zone, the parallelism minimum zone is allowed to float in a direction orthogonal to the datum A surface. For a convex surface, the parallelism error can be computed as: e = max { u\ - u) for i = 1 and y = 1 (22) Fig. 13 Fixture swept cutter surface and the plane equations associated with the displaced workpiece sides (5) map the mathematical parameters that describe the swept cutter surface and the boundaries into the workpiece datum reference frame Having completed these steps, the geometric accuracy of the machined surface can be subsequently assessed. 3.2 Step 2: Geometric Assessment for Known Datum Establishment. In general a surface machined under the presence of datum establishment error will be free of form error. However it will be subject to datum related errors such as position errors, orientation errors, and profile errors. Geometric error can be determined through the formulation of a minimum zone about the predicted machined feature. A minimum zone has the same form, orientation, and position requirements as a tolerance zone with respect to a datum system. However the size of the zone is that at minimum separation such that the upper and/or lower zones contain the extreme geometry of the machined surface. The size of the geometric error is the size of the minimum zone. To illustrate this procedure consider the assessment of the machined surface in Figure 10. Assume that datum feature A is the bottom surface of the workpiece and that datum A is the x-y plane. The nominal machined surface (shown in Figure 8) is parallel to datum A and at a distance of K units. The minimum zone illustrated is that associated with profile error with respect to datum A. Note that the upper and lower zones are equidistant from the nominal surface. The size, e, of the zone is the profile error of the surface. In the case of convex surfaces, the extreme geometry of the surface will always exist at the extreme points. Consequently the profile error can be computed as: e = 2 * max { uf - K\ for i = 1 (21) The assessment of the parallelism error of this surface with 3.3 Step 3: Geometric Assessment for Given Tolerance Scheme. In was shown in section 2, that a given tolerance scheme will result in sixty-four cases of extreme datum establishment error. Each case will have a distinct impact on a particular geometric error. To assess the overall impact of a tolerance scheme on a particular geometric error, all sixty-four cases need to be evaluated and the maximum geometric error identified. Note that if a surface is to be assessed for a variety of geometric errors, there is no guarantee that maxima for each error will occur for the same case of extreme datum establishment error. This will be illustrated in section 5. 4 Simulation Study The locator tolerance analysis model described previously was applied to the fixture-workpiece system shown in Figs. 11, 12, and 13. Two machined surfaces were considered. Each was assessed with respect to angularity error and profile of surface error. The layout of the locators with respect to the workpiece datum features is illustrated in Fig. 14. Note that the clamps hold the part down against the datum A locators and the datum B locators. The part is held against the datum C locator by the frictional forces at the clamp-workpiece contact regions and the datum A-B locator-workpiece contact regions. The objectives of the simulation study were to determine: 1. the sensitivity of the profile and angularity errors with respect to isolated sources of locator variability, 2. the relative impact of a profile tolerance scheme versus a dimensional tolerance scheme on the worst case, profile and angularity errors of the machined surfaces, and 3. the sensitivity of the profile and angularity errors with respect to locator radius. To achieve the first objective, a profile tolerance ranging from 0.002 units to 0.02 units (in increments of 0.002 units) was applied to each locator while the remainder were held at their nominal values. For each case, the locator tolerance analysis procedure was applied to compute the profile and angularity errors of the two machined surfaces. This procedure was repeated for the application of a dimensional tolerance. Machined Surface 1 Locators Machined Surface 1 - Machined Surface 2 Fig. 14 Locator layout 278 / Vol. 121, MAY 1999 Transactions of the ASME

Results of Ranging the Source of Variability 0 Locator 1 X Locator 2 a\ Locator 3 O Locator 4 X Locator 5 O Locator 6 0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140 0.0160 0 01 BO 0.0200 Tolerance on Locator Fig. 15 Geometric error vs. individual locator variation To achieve the second objective, profile tolerances and dimensional tolerances ranging from 0.002 units to 0.02 units (in increments of 0.002 units) were applied to the locators, and the worst case profile and angularity errors of the individual surfaces were computed. Recall that a worst case geometric error is the maximum that results from the set of 64 extreme locator variations. Note all six locators had a nominal radius of 0.25 units. In addition each locator was subject to the same tolerance. To achieve the third objective, the nominal radii of the six locators were varied from 0.25 units to 2.0 units (in increments of 0.02 units). For each case, a profile tolerance of 0.02 units was applied to all six locators, and the worst case geometric errors were computed. This procedure was repeated for the application of a 0.02 unit dimensional tolerance. 5 Results and Discussion The results of ranging the tolerances on the individual locators are typified by Fig. 15. This plot shows the variation of the profile errors of surface 2 as a function of the size of the profile tolerances applied to the individual locators. As can be seen, the relationship is linear. This same linearity was seen for both types of errors on both surfaces, regardless of tolerance scheme. This was surprising in light of the fact that the locator error-datum establishment error model and the machined surface characterization models are nonlinear. This suggests that both models can potentially be linearized with little loss of accuracy. This also supports the basic assumption that the datum related errors take on their greatest values when the geometry's of the locators are allowed to vary to their extremes. The locator variations, which are bounded by individual tolerances, represent independent variables. It is known from linear optimization theory that since a linear relationship exists between these variables and the sizes of the geometric errors, that the maximum error will occur when each of the independent variables is allowed to vary to either its upper or lower bound. The results for all the errors are illustrated in Table 1 for a tolerance of 0.004 units. As can be seen, both the profile errors and angularity errors are sensitive to the location of the source of variability. For example, the profile error of surface 1 is 40 times more sensitive to variations in locator 2 than locator 6. In contrast, the profile error of surface 2 is 70 times more sensitive to variations in locators 4 and 5 than locator 1. Consequently two conclusions can be drawn. The first is that care should be taken with respect to choosing the location of a locator with respect to a datum feature. The second is that the practice of applying a uniform tolerance to all six locators may be unnecessarily restrictive, since the variability of a subset may dominate. The impact of ranging the tolerances simultaneously on the worst case, geometric errors of the machined surfaces are presented in Table 2. As expected, the worst case, geometric errors of both surfaces increase proportionally with increasing locator tolerance. This is shown graphically in Fig. 16 for the profile tolerances. The results also reveal that the size of the profile errors for surfaces 1 and 2 are 4.7 times and 9.7 times as great respectively as the size of the tolerances assigned to the locators. In contrast, the angularity errors of the two surfaces are only 0.9 and 2.3 times as great respectively. This discrepancy is expected since angularity error is affected by rotational datum Table 1 Results of ranging the source of variability Tol. (.004) Tolerance Scheme on Locators Dimensional Tolerance Scheme on Locators on Machined Surface 1 Machined Surface 2 Machined Surface 1 Machined Surface 2 Loc# 1 0.0039 0.0017 0,0002 0.0000 0.0039 0.0017 0.0002 0.0000 2 0.0065 0.0010 0.0044 0.0003 0.0065 0.0010 0.0044 0.0003 3 0.0041 0.0011 0.0037 0.0003 0.0041 0.0011 0.0037 0.0003 4 0.0022 0.0002 0.0135 0.0005 0.0022 0.0002 0.0135 0.0005 5 0.0042 0.0002 0.0139 0.0005 0.0042 0.0002 0.0139 0.0005 6 0.0002 0.0001 0.0041 0.0000 0.0002 0.0001 0.0041 0.0000 Journal of Manufacturing Science and Engineering MAY 1999, Vol. 121 / 279

Table 2 Results of ranging tolerance size simultaneously Tolerance Scheme on Locators Dimensional Tolerance Scheme on Locators Machined Surface 1 Machined Surface 2 Machined Surface 1 Machined Surface 2 Tol. Pronie 0.002 0.0094 0.0018 0.0194 0.0046 0.0094 0.0018 0.0194 0.0046 0.004 0.0188 0.0037 0.0388 0.0093 0.0188 0.0037 0.0388 0.0093 0.006 0.0281 0.0053 0.0582 0.0139 0.0281 0.0053 0.0582 0.0139 0.008 0.0375 0.0071 0.0776 0.0186 0.0375 0.0071 0.0776 0.0186 0.01 0.0469 0.0089 0.0971 0.0232 0.0469 0.0089 0.0971 0.0232 0.012 0.0562 0.0106 0.1166 0.0279 0.0562 0.0106 0.1166 0.0279 0.014 0.0656 0.0131 0.1361 0.0326 0.0656 0.0131 0.1361 0.0326 0.016 0.0750 0.0142 0.1557 0.0373 0.0750 0.0142 0.1557 0.0373 0.018 0.0844 0.0160 0.1753 0.0420 0.0844 0.0160 0.1753 0.0420 0.02 0.0939 0.0182 0.1950 0.0468 0.0939 0.0182 0.1950 0.0468 establishment error only. In contrast profile error is affected by both rotational and translational datum establishment errors. The results also reveal that surface 2 is roughly twice as sensitive to locator variation as surface 1. Another interesting fact that can be derived from Tables 1 and 2 is that the profile tolerance scheme and the dimensional tolerance scheme provide identical results. This is despite the fact that the volume of the profile tolerance zone is twice that of the dimensional tolerance zone. The reason for this is that the region of potential contact between a locator and the workpiece lies within a small vicinity of the nominal point of contact. Near this region, the tolerance zones for both schemes are nearly equivalent, differing only by radius. However as will be discussed shortly, both the profile errors and angularity errors are insensitive to locator radius. Consequently the net effect of each tolerance is the same. In general the dimensional tolerancing scheme places unnecessary control over locator geometry that does not affect datum establishment error. Assuming that the cost of locator manufacture is directly related to allowable variability, it can be concluded that the profile tolerance scheme is superior to the dimensional tolerance scheme. The results of ranging the nominal radius of the locators are presented in Table 3. In general both the profile errors and angularity errors are insensitive to changes in nominal radius. This explains the equivalency in results of the profile and dimensional tolerance schemes. Therefore, from the standpoint of the impact of locator variability on datum establishment error, the choice of locator radius is irrelevant. Alternatively, it has been shown by Shawki and Abdel-aal (1966) and Hockenberger and De Meter (1995,1996) that there is a direct relationship between the nominal radius of a spherical tip locator and the magnitude of deformation that takes place at the workpiece-locator contact region during clamping. In general, the larger the radius of the spherical tip locator, the stiffer the region, and the smaller the amount of deformation and resultant workpiece displacement during clamping. Consequently this suggests that since locator radius has no affect on geometric errors due to locator variability, larger locator radii are preferred. However this statement must be tempered with the fact that in real life, datum features are subject to surface irregularities. As a consequence, without taking into account potential interactions between locator radius and datum feature irregularities, it would be premature to draw conclusions regarding what the best practice is with regard to locator radii selection. In addition, it must be remembered that the largest possible radius is infinite, implying a planar tip locator. A planar tip locator is designed to make planar contact with a planar datum surface as opposed to point contact. Since the models presented in this paper do not cover this possibility, it is not possible to state that planar tip locators are superior to spherical tip locators. 6 Conclusions A common rule of thumb in fixture design is that locator variability should not contribute to more than 10 percent of the allowable variability of a machined surface. To do so, a tool designer must select an adequate tolerance scheme. However this paper has illustrated that the relationship between a locator tolerance scheme and the geometric errors that may result is quite complex. Consequently tolerance allocation without the benefit of analytical verification is potentially risky. This paper has presented a methodology for modeling and analyzing the impact of a locator tolerance scheme on the datum Results of Ranging Tolerance Size - Surface 1 Prof. EIT. ~Surfacel Aug. Err. - Surface2 Prof. Err. "Surfacc2 Ane. Err. 0.0000 ' ' I I I 0 000 0.002 0.004 0.006 0.008 0.010 0012 0.014 0.016 0 018 0.020 Tolerance on Locators Fig. 16 Worst case, geometric error vs. simultaneous locator variation 280 / Vol. 121, MAY 1999 Transactions of the ASME

Table 3 Results of ranging nominal locator radius Tolerance Scheme on Locators (0.02) Dimensional Tolerance Scheme on Locators (0.02) Machined Surface 1 Machined Surface 2 Machined Surface 1 Machined Surface 2 radoi 0.25 0.0939 0.0183 0.1950 0.0468 0.0939 0.0183 0.1950 0.0468 0.50 0.0939 0.0183 0.1950 0.0468 0.0939 0.0183 0.1950 0.0468 0.75 0.0939 0.0183 0.1950 0.0468 0.0939 0.0183 0.1950 0.0468 1.00 0.0939 0.0183 0.1950 0.0468 0.0939 0.0183 0.1950 0.0468 1.25 0.0939 0.0183 0.1950 0.0468 0.0939 0.0183 0.1950 0.0468 1.50 0.0939 0.0183 0.1950 0.0468 0.0939 0.0183 0.1950 0.0468 1.75 0.0939 0.0183 0.1950 0.0468 0.0939 0.0183 0.1950 0.0468 2.00 0.0939 0.0183 0.1950 0.0468 0.0939 0.0183 0.1950 0.0468 related, geometric errors of linear, machined features. In addition, the evidence gained from the application of this model to a specific simulation study suggests the following: 1. the relationship between locator tolerance size and resultant datum related, geometric error is predominantly linear, 2. datum related, geometric error due to locator variability is insensitive to locator radius, 3. due to conclusion 2, it is best to select larger locator radii in order to minimize the impact of contact region deformation on workpiece displacement during clamping; however this argument can not be extended to planar tip locators at this time since the models presented in this paper do not cover the possibility of planar contact between the locator and workpiece, 4. similar datum related, geometric errors on different surfaces have different sensitivities to locator tolerance size, 5. different datum related, geometric errors have different sensitivities to locator tolerance size, 6. datum related, geometric errors are sensitive to the source of locator variability (i.e. locator position), 7. the practice of applying a uniform tolerance to all six locators may be unnecessarily restrictive since the variability of a subset may dominate, and 8. a profile tolerance scheme and dimensional tolerance scheme will result in the same the potential datum related, geometric errors despite the fact that a profile tolerance is substantially more restrictive. The models developed thus far have been limited to profile and dimensional tolerances applied to spherical tip locators in contact with planar workpiece datum features. Future work will be devoted to expanding these models to handle different locator geometry's and nonlinear datum features and nonlinear machined features. In addition, effort will be devoted to developing models that take into account the combined affects of locator geometric variability, datum feature variability, and contact region deformation. Acknowledgments The authors wish to acknowledge the support provided by the Machine Tool-Agile Manufacturing Research Institute. References American National Standards Institute, 1995, Y 14.5M-1994 Dimensioning and Tolerancing, ASME, New York, New York. Bourdet, P., and Clement, A., 1974, "Optimalisation des Montages d'usinage," L'lngenieur et le Techniciien de VEnseignement Technique - 7/8-74. Bourdet, P., and Clement, A., 1988, "A Study of Optimal-Criteria Identification Based on the Small Displacement Screw Model," Annals of the CIRP, Vol. 37/ 1/1988, pp. 503-506. Hockenberger, M. J., and De Meter, E. C, 1995, "The Effect of Machining Fixture Design Parameters on Workpiece Displacement," Manufacturing Review, Vol. 8, No. 1, pp. 22-31. Hockenberger, M. J., and De Meter, E. C, 1996, "The Application of Meta Functions to the Quasi-Static Analysis of Workpiece Displacement Within a Machining Fixture," ASME JOURNAL OF MANUFACTURING SCIENCE AND ENGI NEERING, Vol. 118, No. 3, pp. 325-331. Laloum, M., Dar-El, I., and Weill, R., 1989, "Optimizing Positioning of Mechanical Parts," The International Conference on CAD/CAM and AMT Binyanei Ha'Ooma, Jerusualem, pp. 26-32. Shawki, G. S., and Abdel-Aal, 1966, "Rigidity Considerations in Fixture Design Contact Rigidity at Locating Elements," International Journal of Machine Tool Design and Research, Vol. 6, pp. 31-43. Weill, R., Dar-El, I., and Laloum, M., 1991, "The Influence of Fixture Positioning s on the Geometric Accuracy of Mechanical Parts," Proceedings of the CIRP Conference on PE & MS, pp. 215-225. Journal of Manufacturing Science and Engineering MAY 1999, Vol. 121 / 281