**4. Geometrical tolerancing in reverse engineering**

In order to observe interchangeability, *geometrical* as well as dimensional accuracy specifications of an RE component must comply with those of the mating part(-s). GD&T in RE must ensure that a reconstructed component will fit and perform well without affecting the function of the specific assembly. The methodology that is presented in this section focuses on the RE assignment of the main type of geometrical tolerance that is used in industry, due to its versatility and economic advantages, the True Position tolerance. However, the approach can be easily adapted for RE assignment of other location geometrical tolerances types, such as coaxiality or symmetry and, as well as, for run-out or profile tolerances.

Position tolerancing is standardized in current GD&T international and national standards, such as (ISO, 1998; ISO 1101, 2004; ASME, 2009). Although the ISO and the ASME

geometric counterpart of the mating features with reference to one, two or three Cartesian datums. The relationship between mating features in such a clearance fit may be classified either as a fixed or a floating fastener type, (ASME, 2009; Drake, 1999), Figure 3. Floating fastener situation exists where two or more parts are assembled with fasteners such as bolts and nuts, and all parts have clearance holes for the bolts. In a fixed fastener situation one or more of the parts to be assembled have clearance holes and the mating part has restrained fasteners, such as screws in threaded holes or studs. The approach that is here presented deals with *both* the floating and fixed fastener cases by integrating the individual case

Fig. 3. Typical floating and fixed fasteners and worst case assembly conditions (Drake, 1999)

Basic issues of the assignment of a Position Tolerance in RE are included in Table 1. Limited number of reference components that does not allow for statistical analysis, availability or not of the mating parts and the presence of wear may affect the reliability of the RE results. Moreover, datum selection and the size of the position tolerance itself should ensure, obviously, a stress-free mechanical mating. The analytic approach presented in this section deals with the full range of these issues in order to produce a reliable solution within

*i. The number of available RE components that will be measured. The more RE parts are measured, the more reliable will be the extracted results. Typically, the number of available RE components is extremely limited, usually ranging from less than ten to a single one article. ii. Off the shelf, worn or damaged RE components. Off the shelf RE components are obviously ideal for the job, given that the extent of wear or damage is for the majority of cases difficult to* 

*iv. Existence of repetitive features in the RE component that may have the same function (group* 

*vi. The size and the form (cylindrical, circular, square, other) of the geometrical tolerance zone. vii. Candidate datums and datum reference frames. Depending on the case more possible DRFs* 

*x. Assignment of Maximum Material and Least Material Conditions to both the RE-feature and* 

*xi. Measurement instrumentation capabilities in terms of final uncertainty of the measurements* 

methodologies published by (Kaisarlis et al. 2007; 2008).

realistic time.

*be quantified or compensated. iii. Accessibility of the mating part (-s).* 

*viii. Precedence of datum features in DRFs. ix. Theoretical (basic) dimensions involved.* 

*v. Type of assembly (e.g. floating or fixed fasteners).* 

*results. Measurements methods and software.* 

Table 1. Issues of Geometrical Tolerance Assignment in RE

*or pattern of features).* 

*may be considered.* 

*RE datum features.* 

tolerancing systems are not fully compatible, they both define position geometrical tolerance as the total permissible variation in the location of a feature about its exact true position. For cylindrical features such as holes or bosses the position tolerance zone is usually the diameter of the cylinder within which the axis of the feature must lie, the center of the tolerance zone being at the exact true position, Figure 2, whereas for size features such as slots or tabs, it is the total width of the tolerance zone within which the center plane of the feature must lie, the center plane of the zone being at the exact true position. The position tolerance of a feature is denoted with the size of the diameter of the cylindrical tolerance zone (or the distance between the parallel planes of the tolerance zone) in conjunction with the theoretically exact dimensions that determine the true position and their relevant datums, Figure 2. Datums are, consequently, fundamental building blocks of a positional tolerance frame in positional tolerancing. Datum features are chosen to position the toleranced feature in relation to a Cartesian system of three mutually perpendicular planes, jointly called Datum Reference Frame (DRF), and restrict its motion in relation to it. Positional tolerances often require a three plane datum system, named as primary, secondary and tertiary datum planes. The required number of datums (1, 2 or 3) is derived by considering the degrees of freedom of the toleranced feature that need to be restricted. Change of the datums and/or their order of precedence in the DRF results to different geometrical accuracies, (Kaisarlis et al., 2008).

Fig. 2. Cylindrical tolerance zone and geometric true position tolerancing for a cylindrical feature according to ISO 1101 (Kaisarlis et al, 2008)

The versatility and economic benefits of true position tolerances are particularly enhanced when they are assigned at the Maximum Material Condition (MMC). At MMC, an increase in position tolerance is allowed, equal to the departure of the feature from the maximum material condition size, (ISO, 1988; ASME, 2009). As a consequence, a feature with size beyond maximum material but within the dimensional tolerance zone and its axis lying inside the enlarged MMC cylinder is acceptable. The accuracy required by a position tolerance is thus relaxed through the MMC assignment and the reject rate reduced. Moreover, according to the current ISO and ASME standards, datum features of size that are included in the DRF of position tolerances can also apply on either MMC, Regardless of Feature Size (RFS) or Least Material Condition (LMC) basis.

Position tolerances mainly concern clearance fits. They achieve the intended function of a clearance fit by means of the relative positioning and orientation of the axis of the true geometric counterpart of the mating features with reference to one, two or three Cartesian datums. The relationship between mating features in such a clearance fit may be classified either as a fixed or a floating fastener type, (ASME, 2009; Drake, 1999), Figure 3. Floating fastener situation exists where two or more parts are assembled with fasteners such as bolts and nuts, and all parts have clearance holes for the bolts. In a fixed fastener situation one or more of the parts to be assembled have clearance holes and the mating part has restrained fasteners, such as screws in threaded holes or studs. The approach that is here presented deals with *both* the floating and fixed fastener cases by integrating the individual case methodologies published by (Kaisarlis et al. 2007; 2008).

Fig. 3. Typical floating and fixed fasteners and worst case assembly conditions (Drake, 1999)

Basic issues of the assignment of a Position Tolerance in RE are included in Table 1. Limited number of reference components that does not allow for statistical analysis, availability or not of the mating parts and the presence of wear may affect the reliability of the RE results. Moreover, datum selection and the size of the position tolerance itself should ensure, obviously, a stress-free mechanical mating. The analytic approach presented in this section deals with the full range of these issues in order to produce a reliable solution within realistic time.


140 Reverse Engineering – Recent Advances and Applications

tolerancing systems are not fully compatible, they both define position geometrical tolerance as the total permissible variation in the location of a feature about its exact true position. For cylindrical features such as holes or bosses the position tolerance zone is usually the diameter of the cylinder within which the axis of the feature must lie, the center of the tolerance zone being at the exact true position, Figure 2, whereas for size features such as slots or tabs, it is the total width of the tolerance zone within which the center plane of the feature must lie, the center plane of the zone being at the exact true position. The position tolerance of a feature is denoted with the size of the diameter of the cylindrical tolerance zone (or the distance between the parallel planes of the tolerance zone) in conjunction with the theoretically exact dimensions that determine the true position and their relevant datums, Figure 2. Datums are, consequently, fundamental building blocks of a positional tolerance frame in positional tolerancing. Datum features are chosen to position the toleranced feature in relation to a Cartesian system of three mutually perpendicular planes, jointly called Datum Reference Frame (DRF), and restrict its motion in relation to it. Positional tolerances often require a three plane datum system, named as primary, secondary and tertiary datum planes. The required number of datums (1, 2 or 3) is derived by considering the degrees of freedom of the toleranced feature that need to be restricted. Change of the datums and/or their order of precedence in the DRF results to different

Fig. 2. Cylindrical tolerance zone and geometric true position tolerancing for a cylindrical

The versatility and economic benefits of true position tolerances are particularly enhanced when they are assigned at the Maximum Material Condition (MMC). At MMC, an increase in position tolerance is allowed, equal to the departure of the feature from the maximum material condition size, (ISO, 1988; ASME, 2009). As a consequence, a feature with size beyond maximum material but within the dimensional tolerance zone and its axis lying inside the enlarged MMC cylinder is acceptable. The accuracy required by a position tolerance is thus relaxed through the MMC assignment and the reject rate reduced. Moreover, according to the current ISO and ASME standards, datum features of size that are included in the DRF of position tolerances can also apply on either MMC, Regardless of

Position tolerances mainly concern clearance fits. They achieve the intended function of a clearance fit by means of the relative positioning and orientation of the axis of the true

geometrical accuracies, (Kaisarlis et al., 2008).

feature according to ISO 1101 (Kaisarlis et al, 2008)

Feature Size (RFS) or Least Material Condition (LMC) basis.


Table 1. Issues of Geometrical Tolerance Assignment in RE

For the fixed fasteners case, in industrial practice the *total position tolerance* TPOS of equation (16-i) is distributed between shaft and hole according to the ease of manufacturing, production restrictions and other factors that influence the manufacturing cost of the mating parts. In conformance with that practice a set of 9 candidate sizes for the position tolerance of the RE-shaft, *RCAN\_s* and/ or the RE-hole, *RCAN\_h*, is created by the method with a (TPOS

*RCAN\_h* = *RCAN\_s* = {TPOS1 , TPOS2 ,…, TPOSi ,…,TPOS9}

TPOSi\_MAX = TPOSi + LMCh – MMCh (RE-feature / Hole)

To ensure proper RE-part interfacing and safeguard repeatability, datum features of the original part and those of the RE-part should, ideally, coincide. In order to observe this principle the original datum features and their order of precedence have to be determined. Initial recognition of datum features among the features of the RE-part is performed interactively following long established design criteria for locating or functional surfaces and the same, and taking into consideration the mating parts function. Out of all candidate recognized datums an initial set of candidate DRFs, *DCAN\_INIT*, is produced by taking all combinations in couples and in triads between them. A valid DRF should conform with the constraints that have to do with the arrangement and the geometrical deviations of its datums. Only DRFs that arrest all degrees of freedom of the particular RE-feature and consequently have three or at least two datums are considered. DRF qualification for geometric feasibility is verified by reference to the list of the valid geometrical relationships between datums as given in (ASME, 1994). The geometric relationship for instance, for the usual case of three datum planes that construct a candidate DRF is in this way validated, i.e. the primary datum not to be parallel to the secondary and the plane used as tertiary datum not to be parallel to the line constructed by the intersection of the primary and secondary datum planes. Planar or axial datum features are only considered by the method as primary when the axis of the RE-feature is perpendicular in the first case or parallel, in the second

The following analysis applies for both the hole and the shaft and is common for the fixed and floating fasteners case. Consequently, the indexes "*h"* or "*s"* are not used hereafter. It is here also noted that the index "*i"* only concerns the fixed fastener case. For the floating fastener case the index "*i"* has a constant value of 1. Let *RFDF* be the set of the measured form deviations of a candidate datum feature and *RO* the orientation deviations of the RE feature axis of symmetry with respect to that datum. Fitness of the members of the initial DRF set, *DCAN\_INIT*, against the members of the *RCAN* set of candidate position tolerance sizes

is confirmed regarding the primary datum through the following constraints,

TPOSi\_MAX = TPOSi + MMCs – LMCs (RE- feature / Shaft) (18)

For the floating fasteners case the *total position tolerance* TPOS of equation (16-ii) actually concerns only RE-features of the Hole type. Therefore, the *RCAN\_h* set only contains the TPOS element. The above tolerances attain, apparently, their maximum values when the RE

where, TPOSi = i ·TPOS /10, i=1, 2, ..., 9 (17)

/10**)** step, which includes the 50% -50% case,

feature own dimensional tolerance zone is added,

one, to them.

**4.2 Sets of candidate DRFs and theoretical dimensions** 

Assignment of RE-position tolerance for both the fixed and the floating fastener case is accomplished by the present method in five sequential steps. The analysis is performed individually for each feature that has to be toleranced in the RE-component. At least two RE reference components, intact or with negligible wear, need to be available in order to minimize the risk of measuring a possibly defective or wrongly referenced RE component and, as it is later explained in this section, to improve the method efficiency. This does not certainly mean that the method cannot be used even when only one component is available. Mating part availability is desirable as it makes easier the datum(s) recognition. Minimum assembly clearance and, as well as, the dimensional tolerance of the RE-feature *(hole, peg, pin or screw shaft)* and RE-Datums *(for features of size)* are taken as results from the RE dimensional tolerance analysis presented in the previous section of the chapter in conjunction with those quoted in relevant application- specific standards.

The primary step (a) of the analysis concerns the recognition of the critical features on the RE component that need to be toleranced and, as well as, their fastening situation. This step is performed interactively and further directs the analysis on either the fixed or the floating fastener option. In step (b) mathematical relationships that represent the geometric constraints of the problem are formulated. They are used for the establishment of an initial *set of candidate position tolerances.* The next step (c) qualifies *suggested sets* out of the group (b) that have to be in conformance with the measured data of the particular RE-feature. The step (d) of the analysis produces then a set of *preferred position tolerances* by filtering out the output of step (c) by means of knowledge-based rules and/or guidelines. The capabilities and expertise of the particular machine shop, where the new components will be produced, and the cost-tolerance relationship, are taken into consideration in the last step (e) of the analysis, where the required position tolerance is finally obtained. For every *datum feature* that can be considered for the position tolerance assignment of an *RE*-*feature*, the input for the analysis consists of *(i)* the measured form deviation of the datum feature (e.g. flatness), *(ii)* its measured size, in case that the datum is a feature of size (e.g. diameter of a hole) and *(iii)* the orientation deviation (e.g. perpendicularity) of the RE-feature axis of symmetry with respect to that datum. The orientation deviations of the latter with respect to the two other datums of the same DRF have also to be included (perpendicularity, parallelism, angularity). Input data relevant with the RE-feature itself include its measured size (e.g. diameter) and coordinates, e.g. X, Y measured dimensions by a CMM, that locate its axis of symmetry. Uncertainty of the measured data should conform to the pursued accuracy level. In that context the instrumentation used for the measured input data, e.g. ISO 10360-2 accuracy threshold for CMMs, is considered appropriate for the analysis only if its uncertainty is at six times less than the minimum assembly clearance.

#### **4.1 Sets of candidate position tolerance sizes**

The size of the total position tolerance zone is determined by the minimum clearance, min*CL*, of the (hole, screw-shaft) assembly. It ensures that mating features will assemble even at worst case scenario, i.e. when both parts are at MMC and located at the extreme ends of the position tolerance zone (ASME, 2009). The equations (16 -i) and (16-ii) apply for the fixed and floating fastener case respectively,

$$\begin{aligned} \text{(i)} \quad \text{T}\_{\text{POS}} &= \text{minCL} = \text{T}\_{\text{PCS}\_{\text{-}}s} + \text{T}\_{\text{PCS}\_{\text{-}}h} = \text{MMC}\_{\text{h}} - \text{MMC}\_{\text{s}}\\ \text{(ii)} \quad \text{T}\_{\text{PCS}} &= \text{minCL} = \text{T}\_{\text{PCS}\_{\text{-}}h} \end{aligned} \tag{16}$$

For the fixed fasteners case, in industrial practice the *total position tolerance* TPOS of equation (16-i) is distributed between shaft and hole according to the ease of manufacturing, production restrictions and other factors that influence the manufacturing cost of the mating parts. In conformance with that practice a set of 9 candidate sizes for the position tolerance of the RE-shaft, *RCAN\_s* and/ or the RE-hole, *RCAN\_h*, is created by the method with a (TPOS /10**)** step, which includes the 50% -50% case,

$$\begin{array}{l} \mathbf{R}\_{\text{CAN},h} = \mathbf{R}\_{\text{CAN},s} = \{ \mathbf{T}\_{\text{POS1}}, \mathbf{T}\_{\text{PCS2}}, \dots, \mathbf{T}\_{\text{PCSi}}, \dots, \mathbf{T}\_{\text{PCS}} \} \\\text{where} \quad \mathbf{T}\_{\text{PCSi}} = \mathbf{i} \cdot \mathbf{T}\_{\text{PCS}} / 10, \qquad \mathbf{i} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{9} \end{array} \tag{17}$$

For the floating fasteners case the *total position tolerance* TPOS of equation (16-ii) actually concerns only RE-features of the Hole type. Therefore, the *RCAN\_h* set only contains the TPOS element. The above tolerances attain, apparently, their maximum values when the RE feature own dimensional tolerance zone is added,

$$\begin{array}{ll} \text{T}\_{\text{PCSi\\_MAX}} = \text{T}\_{\text{PCSi}} + \text{LMC}\_{\text{h}} - \text{MMC}\_{\text{h}} & \text{(RE-feature / Hole)}\\ \text{T}\_{\text{PCSi\\_MAX}} = \text{T}\_{\text{PCSi}} + \text{MMC}\_{\text{s}} - \text{LMC}\_{\text{s}} & \text{(RE-feature / Shaft)} \end{array} \tag{18}$$

#### **4.2 Sets of candidate DRFs and theoretical dimensions**

142 Reverse Engineering – Recent Advances and Applications

Assignment of RE-position tolerance for both the fixed and the floating fastener case is accomplished by the present method in five sequential steps. The analysis is performed individually for each feature that has to be toleranced in the RE-component. At least two RE reference components, intact or with negligible wear, need to be available in order to minimize the risk of measuring a possibly defective or wrongly referenced RE component and, as it is later explained in this section, to improve the method efficiency. This does not certainly mean that the method cannot be used even when only one component is available. Mating part availability is desirable as it makes easier the datum(s) recognition. Minimum assembly clearance and, as well as, the dimensional tolerance of the RE-feature *(hole, peg, pin or screw shaft)* and RE-Datums *(for features of size)* are taken as results from the RE dimensional tolerance analysis presented in the previous section of the chapter in

The primary step (a) of the analysis concerns the recognition of the critical features on the RE component that need to be toleranced and, as well as, their fastening situation. This step is performed interactively and further directs the analysis on either the fixed or the floating fastener option. In step (b) mathematical relationships that represent the geometric constraints of the problem are formulated. They are used for the establishment of an initial *set of candidate position tolerances.* The next step (c) qualifies *suggested sets* out of the group (b) that have to be in conformance with the measured data of the particular RE-feature. The step (d) of the analysis produces then a set of *preferred position tolerances* by filtering out the output of step (c) by means of knowledge-based rules and/or guidelines. The capabilities and expertise of the particular machine shop, where the new components will be produced, and the cost-tolerance relationship, are taken into consideration in the last step (e) of the analysis, where the required position tolerance is finally obtained. For every *datum feature* that can be considered for the position tolerance assignment of an *RE*-*feature*, the input for the analysis consists of *(i)* the measured form deviation of the datum feature (e.g. flatness), *(ii)* its measured size, in case that the datum is a feature of size (e.g. diameter of a hole) and *(iii)* the orientation deviation (e.g. perpendicularity) of the RE-feature axis of symmetry with respect to that datum. The orientation deviations of the latter with respect to the two other datums of the same DRF have also to be included (perpendicularity, parallelism, angularity). Input data relevant with the RE-feature itself include its measured size (e.g. diameter) and coordinates, e.g. X, Y measured dimensions by a CMM, that locate its axis of symmetry. Uncertainty of the measured data should conform to the pursued accuracy level. In that context the instrumentation used for the measured input data, e.g. ISO 10360-2 accuracy threshold for CMMs, is considered appropriate for the analysis only if its

conjunction with those quoted in relevant application- specific standards.

uncertainty is at six times less than the minimum assembly clearance.

The size of the total position tolerance zone is determined by the minimum clearance, min*CL*, of the (hole, screw-shaft) assembly. It ensures that mating features will assemble even at worst case scenario, i.e. when both parts are at MMC and located at the extreme ends of the position tolerance zone (ASME, 2009). The equations (16 -i) and (16-ii) apply for

> *(i)* TPOS = minCL =TPOS\_s + TPOS\_h = MMCh – MMCs *(ii)* TPOS = minCL = TPOS\_h

(16)

**4.1 Sets of candidate position tolerance sizes** 

the fixed and floating fastener case respectively,

To ensure proper RE-part interfacing and safeguard repeatability, datum features of the original part and those of the RE-part should, ideally, coincide. In order to observe this principle the original datum features and their order of precedence have to be determined. Initial recognition of datum features among the features of the RE-part is performed interactively following long established design criteria for locating or functional surfaces and the same, and taking into consideration the mating parts function. Out of all candidate recognized datums an initial set of candidate DRFs, *DCAN\_INIT*, is produced by taking all combinations in couples and in triads between them. A valid DRF should conform with the constraints that have to do with the arrangement and the geometrical deviations of its datums. Only DRFs that arrest all degrees of freedom of the particular RE-feature and consequently have three or at least two datums are considered. DRF qualification for geometric feasibility is verified by reference to the list of the valid geometrical relationships between datums as given in (ASME, 1994). The geometric relationship for instance, for the usual case of three datum planes that construct a candidate DRF is in this way validated, i.e. the primary datum not to be parallel to the secondary and the plane used as tertiary datum not to be parallel to the line constructed by the intersection of the primary and secondary datum planes. Planar or axial datum features are only considered by the method as primary when the axis of the RE-feature is perpendicular in the first case or parallel, in the second one, to them.

The following analysis applies for both the hole and the shaft and is common for the fixed and floating fasteners case. Consequently, the indexes "*h"* or "*s"* are not used hereafter. It is here also noted that the index "*i"* only concerns the fixed fastener case. For the floating fastener case the index "*i"* has a constant value of 1. Let *RFDF* be the set of the measured form deviations of a candidate datum feature and *RO* the orientation deviations of the RE feature axis of symmetry with respect to that datum. Fitness of the members of the initial DRF set, *DCAN\_INIT*, against the members of the *RCAN* set of candidate position tolerance sizes is confirmed regarding the primary datum through the following constraints,

(a) (b)

ΔΥ(ij)M= max(*CCAN (ij)YM*)– min(*CCAN(ij)YM*) (26)

ΔX(ij)M = max(*CCAN(ij)XM*)– min(*CCAN(ij)XM*)

Maximum or Least Material Conditions are applied to RE-feature size, Figure 4(b),

In case constraint (25) is not satisfied, a DRF(i)j can only be further considered, when

 max{ΔX(ij)M, ΔΥ(ij)M} ≤ TPOSi\_MAX (27) In case no member of a *DCAN(i)* (i=1,2,…9) set satisfies either constraint (25) or constraint (27) the relevant TPOSi is excluded from the set of Suggested Position Tolerance Sizes, *RSG*.

Let r be the number of the available RE-parts. Sets of *Suggested Theoretical Dimensions*, [*CSG(ij)X*, *CSG(ij)Y*], can now be filtered out of the Candidate Theoretical Dimensions through

(a) (b) Fig. 5. Qualification conditions for suggested theoretical dimensions (Kaisarlis et al., 2007)

, |Y(ij)k – Y(ij)Mu | ≤

2 POSi\_MAX T

(28)

2 POSi\_MAX T

and the constraint imposed by the geometry of a position tolerance, Figure 5(a),

m=1,2, …,p ; k=1,2,…,q ; u=1,2,…,r

Fig. 4. Qualification conditions for suggested DRFs (Kaisarlis et al., 2007)

where,

the application of the relationships,


$$\max(\mathbf{R}\mathbf{F}\_{\text{D}\mathcal{V}}) \leq \mathbf{T}\_{\text{PCSi}} \tag{19}$$

$$\max(\mathbf{RO}) \le \mathbf{T}\_{\text{PCSI}} \tag{20}$$

Mutual orientation deviations of the secondary and/or tertiary datums, *RODF*, in a valid DRF should also conform with the position tolerance of equation (16),

$$\begin{array}{ll}\texttt{max}(\texttt{RF}\_{\textrm{DF}}) \leq \texttt{k} \cdot \texttt{T}\_{\textrm{FOSi}} & \texttt{max}(\texttt{RO}) \leq \texttt{k} \cdot \texttt{T}\_{\textrm{FCSi}} \\\texttt{max}(\texttt{RO}\_{\textrm{DF}}) \leq \texttt{k} \cdot \texttt{T}\_{\textrm{FOSi}} & \texttt{k} \geq 1 \end{array} \tag{21}$$

where k is a weight coefficient depending on the accuracy level of the case. A set of *Candidate* DRFs is thus created*, DCAN (i),* that is addressed to each i member (i=1,…9) of the *RCAN* set.

Sets of *Candidate Theoretical Dimensions*, *[(CCAN(ij)X, CCAN(ij)Y), i=1,2,…9, j=1,2,…,n]*, which locate the RE feature axis of symmetry with reference to every one of the n candidate DRF(i)j of the *DCAN(i)* set are generated at the next stage of the analysis. Measured, from all the available RE reference parts, *axis location coordinate*s are at first integrated into sets, *[CCAN(ij)XM,* **C***CAN(ij)YM]*. Sets of Candidate Theoretical Dimensions are then produced in successive steps starting from those calculated from the integral part of the difference between the minimum measured coordinates and the size of the position tolerance, TPOSi,

$$\mathbf{X}^{\langle \overline{\mathbf{u}} \rangle\_1} = \text{int}[\min(\mathbf{C}\_{\complement\mathcal{N}}\,\overline{\mathbf{u}})\,\mathrm{X}\_M] - \mathrm{T}\_{\text{FCS}\overline{\mathbf{u}}}\,\mathrm{J}\_{\nu}\mathbf{Y}^{\langle \overline{\mathbf{u}} \rangle\_1} = \text{int}[\min(\mathbf{C}\_{\complement\mathcal{N}}\,\overline{\mathbf{u}})\,\mathrm{Y}\_M] - \mathrm{T}\_{\text{FCS}\overline{\mathbf{u}}}\,\mathrm{J} \tag{22}$$

Following members of the *CCAN (ij)X, CCAN (ij)Y* sets are calculated by an incremental increase δ of the theoretical dimensions X(ij)1, Y(ij)1 ,

X(ij)2 = X(ij)1 + δ, Y(ij)2 = Y(ij)1 + δ X(ij)3 = X(ij)2 + δ, Y(ij)3 = Y(ij)2 + δ …………… …………… X(ij)p = X(ij)(p-1) + δ, Y(ij)q = Y(ij)(q-1) + δ (23)

where as upper limit is taken that of the maximum measured X(ij)M, Y(ij)M coordinates plus the position tolerance TPOSi,

$$\mathsf{X}^{\langle \overline{\eta} \rangle}\_{\mathsf{P}} \le \max \{ \mathsf{C}\_{F}(\overline{\eta}) X\_{M} \} + \mathsf{T}\_{\mathsf{FCS\bar{\eta}}} \quad \mathsf{Y}^{\langle \overline{\eta} \rangle}\_{\mathsf{q}} \le \max \{ \mathsf{C}\_{F}(\overline{\eta}) Y\_{M} \} + \mathsf{T}\_{\mathsf{FCS\bar{\eta}}} \tag{24}$$

with the populations p, q of the produced *CCAN(ij)X and CCAN(ij)Y* sets of candidate theoretical dimensions not necessarily equal. In the case study that is presented δ=0.05mm. Other δvalues can be used as well.

#### **4.3 Sets of suggested position tolerances**

Sets of *Suggested DFR*s that are produced in step (b), *DSG(i)*, are qualified as subgroups of the sets of *Candidate DFR*s, *DCAN(i)*, in accordance with their conformance with the measured location coordinates and the application or not of the Maximum or Least Material Conditions to the RE-feature size or to the RE-Datum size. In conjunction with equation (16), qualification criterion for the Suggested DFR's, DRF(i)j j=1,2,…, n, is, Figure 4(a),

$$\max\{\Delta\mathbf{X}(\overline{\mathbf{u}})\_{\mathcal{M}}, \Delta\mathbf{Y}(\overline{\mathbf{u}})\_{\mathcal{M}}\} \le \mathbf{T}\_{\text{PCS}\overline{\mathbf{u}}}\tag{25}$$

Fig. 4. Qualification conditions for suggested DRFs (Kaisarlis et al., 2007)

where,

(23)

144 Reverse Engineering – Recent Advances and Applications

max(*RFDF*)≤ TPOSi (19)

 max(*RO*)≤ TPOSi (20) Mutual orientation deviations of the secondary and/or tertiary datums, *RODF*, in a valid

where k is a weight coefficient depending on the accuracy level of the case. A set of *Candidate* DRFs is thus created*, DCAN (i),* that is addressed to each i member (i=1,…9) of the

Sets of *Candidate Theoretical Dimensions*, *[(CCAN(ij)X, CCAN(ij)Y), i=1,2,…9, j=1,2,…,n]*, which locate the RE feature axis of symmetry with reference to every one of the n candidate DRF(i)j of the *DCAN(i)* set are generated at the next stage of the analysis. Measured, from all the available RE reference parts, *axis location coordinate*s are at first integrated into sets, *[CCAN(ij)XM,* **C***CAN(ij)YM]*. Sets of Candidate Theoretical Dimensions are then produced in successive steps starting from those calculated from the integral part of the difference between the minimum measured coordinates and the size of the position tolerance, TPOSi,

 X(ij)1 = int[min(*CCAN(ij)XM* ) – TPOSi],Y(ij)1 = int[min(*CCAN(ij)YM*) – TPOSi] (22) Following members of the *CCAN (ij)X, CCAN (ij)Y* sets are calculated by an incremental increase

where as upper limit is taken that of the maximum measured X(ij)M, Y(ij)M coordinates plus

 X(ij)p ≤ max(*CP(ij)XM*)+ TPOSi, Y(ij)q ≤ max(*CP(ij)YM*)+ TPOSi (24) with the populations p, q of the produced *CCAN(ij)X and CCAN(ij)Y* sets of candidate theoretical dimensions not necessarily equal. In the case study that is presented δ=0.05mm. Other δ-

Sets of *Suggested DFR*s that are produced in step (b), *DSG(i)*, are qualified as subgroups of the sets of *Candidate DFR*s, *DCAN(i)*, in accordance with their conformance with the measured location coordinates and the application or not of the Maximum or Least Material Conditions to the RE-feature size or to the RE-Datum size. In conjunction with equation (16),

max{ΔX(ij)M, ΔΥ(ij)M} ≤ TPOSi (25)

qualification criterion for the Suggested DFR's, DRF(i)j j=1,2,…, n, is, Figure 4(a),

X(ij)2 = X(ij)1 + δ, Y(ij)2 = Y(ij)1 + δ X(ij)3 = X(ij)2 + δ, Y(ij)3 = Y(ij)2 + δ …………… …………… X(ij)p = X(ij)(p-1) + δ, Y(ij)q = Y(ij)(q-1) + δ

max(*RFDF* )≤ k·TPOSi , max(*RO*)≤ k·TPOSi (21) max(*RODF*)≤ k·TPOSi , k ≥<sup>1</sup>

DRF should also conform with the position tolerance of equation (16),

*RCAN* set.

δ of the theoretical dimensions X(ij)1, Y(ij)1 ,

the position tolerance TPOSi,

values can be used as well.

**4.3 Sets of suggested position tolerances** 

$$\begin{aligned} \Delta \mathbf{X}^{\langle \overline{\eta} \rangle\_{\mathcal{M}}} &= \max \{ \mathbf{C}\_{\text{CAN}^{\langle \overline{\eta} \rangle}} \mathbf{C}\_{\text{M}} \} - \min \{ \mathbf{C}\_{\text{CAN}^{\langle \overline{\eta} \rangle}} \overline{\mathbf{C}}\_{\text{M}} \} \\ \Delta \mathbf{Y}^{\langle \overline{\eta} \rangle\_{\mathcal{M}}} &= \max \{ \mathbf{C}\_{\text{CAN}^{\langle \overline{\eta} \rangle}} \overline{\mathbf{y}}^{\langle \overline{\eta} \rangle} Y\_{\text{M}} \} - \min \{ \mathbf{C}\_{\text{CAN}^{\langle \overline{\eta} \rangle}} \overline{\mathbf{y}}^{\langle \overline{\eta} \rangle} Y\_{\text{M}} \} \end{aligned} \tag{26}$$

In case constraint (25) is not satisfied, a DRF(i)j can only be further considered, when Maximum or Least Material Conditions are applied to RE-feature size, Figure 4(b),

$$\max\{\Delta\text{X}(\overline{\eta})\_{\text{M}\_{\text{M}}}, \Delta\text{Y}(\overline{\eta})\_{\text{M}}\} \le \text{T}\_{\text{YCSi}, \text{MAX}}\tag{27}$$

In case no member of a *DCAN(i)* (i=1,2,…9) set satisfies either constraint (25) or constraint (27) the relevant TPOSi is excluded from the set of Suggested Position Tolerance Sizes, *RSG*.

Let r be the number of the available RE-parts. Sets of *Suggested Theoretical Dimensions*, [*CSG(ij)X*, *CSG(ij)Y*], can now be filtered out of the Candidate Theoretical Dimensions through the application of the relationships,

$$\mid \text{ X}^{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\langle{\tiny{\langle{}}}}}}}}}}}} - \text{X}^{\tiny{\text{\tiny{\text{P}}}}}} \mid \text{\&}} \mid \text{\&} \frac{\text{T}\_{\text{P}\text{Si}\\_{\text{MAX}}}}{2}}{2}, \quad \mid \text{ Y}^{\langle \text{\"}{\langle \text{\tiny{\langle{\langle{\langle{\}}}}}}} - \text{Y}^{\langle \text{\"}{\langle \text{\langle{}}} \text{\"}{\text{A}} \text{\"}} \text{\;}} \mid \text{\&} \frac{\text{T}\_{\text{P}\text{Si}\\_{\text{MAX}}}}{2}}{2} \tag{28}$$
  $\text{m} = 1, 2, \dots, \text{p} \; ; \text{ \&} \qquad \qquad \qquad \text{k} = 1, 2, \dots, \text{p} \; ; \text{ \&} \qquad \qquad \qquad \text{u} = 1, 2, \dots, \text{r} \; \qquad \qquad \qquad \qquad \qquad \text{\&} \qquad \qquad \qquad \text{\&}$ 

and the constraint imposed by the geometry of a position tolerance, Figure 5(a),



To assign cost optimal tolerances to the new RE-components, that have to be remanufactured, the Tolerance Element (TE) method is introduced. Accuracy cost constitutes a vital issue in production, as tight tolerances always impose additional effort and therefore higher manufacturing costs. Within the frame of further development of the CAD tools, emphasis is recently given on techniques that assign mechanical tolerances in terms not only of quality and functionality but also of minimum manufacturing cost. Cost-tolerance functions, however, are difficult to be adopted in the tolerance optimization process because their coefficients and exponents are case-driven, experimentally obtained, and they may well not be representative of the manufacturing environment where the production will take place. The TE method (Dimitrellou et al., 2007a; 2007c, 2008) overcomes the mentioned inefficiencies as it automatically creates and makes use of appropriate cost-tolerance functions for the assembly chain members under consideration. The latter is accomplished through the introduction of the concept of Tolerance Elements (Dimitrellou et al., 2007b) that are geometric entities with attributes associated with the accuracy cost of the specific

The accuracy cost of a part dimension depends on the process and resources required for the production of this dimension within its tolerance limits. Given the workpiece material and the tolerances, the part geometrical characteristics such as shape, size, internal surfaces, feature details and/or position are taken into consideration for planning the machining operations, programming the machine tools, providing for fixtures, etc. These characteristics have thus a direct impact on the machining cost of the required accuracy and determine, indirectly, its magnitude. A Tolerance Element (TE) is defined either as a 3D form feature of particular shape, size and tolerance, or as a 3D form feature of particular position and tolerance (Dimitrellou et al., 2007a). It incorporates attributes associated with the feature shape, size, position, details and the size ratio of the principal dimensions of the part to which it belongs. For a given manufacturing environment (machine tools, inspection equipment, supporting facilities, available expertise) to each TE belongs one directly related

TEs are classified through a five-class hierarch system, Figure 6. Class level attributes are all machining process related, generic and straightforwardly identifiable in conformance with the existing industrial understanding. In first level, TEs are classified according to the basic geometry of the part to which they belong, i.e. rotational TEs and prismatic TEs. Rotational TEs belong to rotational parts manufactured mainly by turning and boring, while prismatic TEs belong to prismatic parts mainly manufactured by milling. The contribution of the geometrical configuration of the part to the accuracy cost of a TE, is taken into account in the second level through the size ratio of the principal dimensions of the part. In this way TEs are classified as short [L/D ≤3] and long [L/D >3] TEs, following a typical way of



**5. Cost - effective RE-tolerance assignment** 

with this environment cost-tolerance function.

machining environment where the components will be manufactured.

respective planar contact surface.

decimal places are preferable.

$$\left(\mathbf{X}(\overline{\mathbf{u}})\_{\mathbf{m}} - \mathbf{X}(\overline{\mathbf{u}})\_{\mathbf{M}\mathbf{u}}\right)^{2} + \left(\mathbf{Y}(\overline{\mathbf{u}})\_{\mathbf{k}} - \mathbf{Y}(\overline{\mathbf{u}})\_{\mathbf{M}\mathbf{u}}\right)^{2} \preceq (\frac{\mathbf{T}\_{\text{POS}}}{2})^{2}\tag{29}$$
  $\mathbf{m} = 1, 2, \dots, \mathbf{p} \; ; \qquad \mathbf{k} = 1, 2, \dots, \mathbf{q} \; ; \qquad \mathbf{u} = 1, 2, \dots, \mathbf{r}$ 

Candidate Theoretical Dimensions that satisfy the constraints (28) but not the constraint (29) can apparently be further considered in conjunction with constraint (27) when Maximum or Least Material Conditions are used. In these cases they are respectively qualified by the conditions, e.g. for the case of RE-feature /Hole, Figure 5(b),

$$\left(\mathbf{X}^{\langle\overline{\mathbf{u}}\rangle\_{\rm m}} - \mathbf{X}^{\langle\overline{\mathbf{u}}\rangle\_{\rm Mu}}\right)^{2} + \left(\mathbf{Y}^{\langle\overline{\mathbf{u}}\rangle\_{\rm k}} - \mathbf{Y}^{\langle\overline{\mathbf{u}}\rangle\_{\rm Mu}}\right)^{2} \leq \left(\frac{\mathbf{T}\_{\rm PỌSi} + \mathbf{d}\_{\rm Mu} \cdot \mathbf{M}\mathbf{M}\mathbf{C}}{2}\right)^{2} \tag{30}$$

$$\left(\mathbf{Y}|\overline{\mathbf{u}}\_{\mathrm{m}} - \mathbf{X}|\overline{\mathbf{u}}\_{\mathrm{Mu}}\right)^{2} + \left(\mathbf{Y}|\overline{\mathbf{u}}\_{\mathrm{k}} - \mathbf{Y}|\overline{\mathbf{u}}\_{\mathrm{Mu}}\right)^{2} \le \frac{\mathbf{T}\_{\mathrm{POS}} + \mathbf{L}\mathbf{M}\mathbf{C} \cdot \mathbf{d}\_{\mathrm{Mu}}}{2}\tag{31}$$
  $\mathbf{m} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{p} \; ; \qquad \qquad \mathbf{k} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{q} \; ; \qquad \mathbf{u} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{r}$ 

When applicable, the case of MMC or LMC on a RE-Datum feature of size may be also investigated. For that purpose, the size tolerance of the datum, TS\_DF, must be added on the right part of the relationships (27) and (28). In that context, the constraints (30) and (31), e.g.

for the case of RE-feature /Hole - RE-Datum /Hole on MMC, are then formulated as,

$$(\mathsf{X}^{(\overline{\mathsf{u}})}\mathsf{m} - \mathsf{X}^{(\overline{\mathsf{u}})}\mathsf{u}\_{\mathsf{Mu}})^{2} + (\mathsf{Y}^{(\overline{\mathsf{u}})}\mathsf{k} - \mathsf{Y}^{(\overline{\mathsf{u}})}\mathsf{u}\_{\mathsf{Mu}})^{2} \lesssim \frac{\mathsf{T}\_{\text{POSl}} + \mathsf{d}\_{\text{Mu}} \cdot \mathsf{M}\mathsf{M}\mathsf{C} + \mathsf{d}\_{\text{Mu\\_DF}} \cdot \mathsf{M}\mathsf{M}\mathsf{C}\_{\text{DF}}}{2})^{2} \tag{32}$$

$$(\text{X}(\overline{\text{u}})\_{\text{m}} - \text{X}(\overline{\text{u}})\_{\text{Mu}})^2 + (\text{Y}(\overline{\text{y}})\_{\text{k}} - \text{Y}(\overline{\text{y}})\_{\text{Mu}})^2 \le (\frac{\text{T}\_{\text{PO}\text{'}} + \text{LMC} - \text{d}\_{\text{Mu}} + \text{d}\_{\text{Mu}\_{\text{DF}}} - \text{MMC}\_{\text{DF}}}{2})^2 \tag{33}$$

$$\mathbf{m} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{p} \; ; \qquad \mathbf{k} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{q} \; ; \qquad \mathbf{u} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{r} \; ;$$

where dMu\_DF is the measured diameter of the datum on the *u-th* RE-part and MMCDF the MMC size of the RE-Datum.

#### **4.4 Sets of preferred position tolerances**

The next step of the analysis provides for three tolerance selection options and the implementation of manufacturing guidelines for datum designation in order the method to propose a limited number of *Preferred Position Tolerance Sets* out of the *Suggested* ones and hence lead the final decision to a rational end result. The first tolerance selection option is only applicable in the fixed fasteners case and focuses for a maximum tolerance size of a TPOS/2. The total position tolerance TPOS, whose distribution between the mating parts is unknown, will be unlikely to be exceeded in this way and therefore, even in the most unfavourable assembly conditions interference will not occur. The second selection option gives its preference to Position Tolerance Sets that are qualified regardless of the application of the Maximum or Least Material Conditions to the RE-feature size and/ or the RE- datum feature size i.e. through their conformance only with the constraint (29) and not the constraints (30) to (33). Moreover, guidelines for datums which are used in the above context are, (ASME 2009; Fischer, 2004):



146 Reverse Engineering – Recent Advances and Applications

Candidate Theoretical Dimensions that satisfy the constraints (28) but not the constraint (29) can apparently be further considered in conjunction with constraint (27) when Maximum or Least Material Conditions are used. In these cases they are respectively qualified by the

When applicable, the case of MMC or LMC on a RE-Datum feature of size may be also investigated. For that purpose, the size tolerance of the datum, TS\_DF, must be added on the right part of the relationships (27) and (28). In that context, the constraints (30) and (31), e.g.

for the case of RE-feature /Hole - RE-Datum /Hole on MMC, are then formulated as,

(X(ij)m – X(ij)Mu ) 2 + (Υ(ij)k – Υ(ij)Mu ) 2 ≤ POSi Mu Mu\_DF DF <sup>2</sup>

(X(ij)m – X(ij)Mu ) 2 + (Υ(ij)k – Υ(ij)Mu ) 2 ≤ POSi Mu Mu\_DF DF <sup>2</sup>

where dMu\_DF is the measured diameter of the datum on the *u-th* RE-part and MMCDF the

The next step of the analysis provides for three tolerance selection options and the implementation of manufacturing guidelines for datum designation in order the method to propose a limited number of *Preferred Position Tolerance Sets* out of the *Suggested* ones and hence lead the final decision to a rational end result. The first tolerance selection option is only applicable in the fixed fasteners case and focuses for a maximum tolerance size of a TPOS/2. The total position tolerance TPOS, whose distribution between the mating parts is unknown, will be unlikely to be exceeded in this way and therefore, even in the most unfavourable assembly conditions interference will not occur. The second selection option gives its preference to Position Tolerance Sets that are qualified regardless of the application of the Maximum or Least Material Conditions to the RE-feature size and/ or the RE- datum feature size i.e. through their conformance only with the constraint (29) and not the constraints (30) to (33). Moreover, guidelines for datums which are used in the above


) <sup>2</sup> T

) <sup>2</sup>

) <sup>2</sup>

( (32)

( (33)

( (30)

( (31)

) <sup>2</sup>

) <sup>2</sup>

MMC - d+ T

d - LMC+ T

MMC - d+MMC - d+ T

MMC - d+d - LMC+ T

( (29)

(X(ij)m – X(ij)Mu ) 2 + (Υ(ij)k – Υ(ij)Mu ) 2 ≤ POSi <sup>2</sup>

(X(ij)m – X(ij)Mu ) 2 + (Υ(ij)k – Υ(ij)Mu ) 2 ≤ POSi Mu <sup>2</sup>

(X(ij)m – X(ij)Mu ) 2 + (Υ(ij)k – Υ(ij)Mu ) 2 ≤ POSi Mu <sup>2</sup>

m=1,2, …,p ; k=1,2,…,q ; u=1,2,…,r

conditions, e.g. for the case of RE-feature /Hole, Figure 5(b),

m=1,2, …,p ; k=1,2,…,q ; u=1,2,…,r

m=1,2, …,p ; k=1,2,…,q ; u=1,2,…,r

**4.4 Sets of preferred position tolerances** 

context are, (ASME 2009; Fischer, 2004):

to prevent tolerances from stacking up excessively

MMC size of the RE-Datum.


### **5. Cost - effective RE-tolerance assignment**

To assign cost optimal tolerances to the new RE-components, that have to be remanufactured, the Tolerance Element (TE) method is introduced. Accuracy cost constitutes a vital issue in production, as tight tolerances always impose additional effort and therefore higher manufacturing costs. Within the frame of further development of the CAD tools, emphasis is recently given on techniques that assign mechanical tolerances in terms not only of quality and functionality but also of minimum manufacturing cost. Cost-tolerance functions, however, are difficult to be adopted in the tolerance optimization process because their coefficients and exponents are case-driven, experimentally obtained, and they may well not be representative of the manufacturing environment where the production will take place. The TE method (Dimitrellou et al., 2007a; 2007c, 2008) overcomes the mentioned inefficiencies as it automatically creates and makes use of appropriate cost-tolerance functions for the assembly chain members under consideration. The latter is accomplished through the introduction of the concept of Tolerance Elements (Dimitrellou et al., 2007b) that are geometric entities with attributes associated with the accuracy cost of the specific machining environment where the components will be manufactured.

The accuracy cost of a part dimension depends on the process and resources required for the production of this dimension within its tolerance limits. Given the workpiece material and the tolerances, the part geometrical characteristics such as shape, size, internal surfaces, feature details and/or position are taken into consideration for planning the machining operations, programming the machine tools, providing for fixtures, etc. These characteristics have thus a direct impact on the machining cost of the required accuracy and determine, indirectly, its magnitude. A Tolerance Element (TE) is defined either as a 3D form feature of particular shape, size and tolerance, or as a 3D form feature of particular position and tolerance (Dimitrellou et al., 2007a). It incorporates attributes associated with the feature shape, size, position, details and the size ratio of the principal dimensions of the part to which it belongs. For a given manufacturing environment (machine tools, inspection equipment, supporting facilities, available expertise) to each TE belongs one directly related with this environment cost-tolerance function.

TEs are classified through a five-class hierarch system, Figure 6. Class level attributes are all machining process related, generic and straightforwardly identifiable in conformance with the existing industrial understanding. In first level, TEs are classified according to the basic geometry of the part to which they belong, i.e. rotational TEs and prismatic TEs. Rotational TEs belong to rotational parts manufactured mainly by turning and boring, while prismatic TEs belong to prismatic parts mainly manufactured by milling. The contribution of the geometrical configuration of the part to the accuracy cost of a TE, is taken into account in the second level through the size ratio of the principal dimensions of the part. In this way TEs are classified as short [L/D ≤3] and long [L/D >3] TEs, following a typical way of

where t0 is the tolerance of the end-dimension D0. For an (i+j)-member dimensional chain dimensions Di constitute the positive members of the chain while dimensions Dj constitute its negative members. In RE tolerancing, preferred alternatives for nominal sizes and dimensional tolerances that are generated from the analysis of Section 3, for each dimension involved in the chain are further filtered out by taking into consideration the above

A second sorting out is applied by taking into account the accuracy cost for each combination of alternatives that obtained in the previous stage. Cost-tolerance functions are

> *n n <sup>k</sup> total i i ii i i C Ct A B t*

where C(t) is the relative cost for the production of the machining tolerance ±t and A, B, k are constants. The combination of alternatives that corresponds to the minimum cost is

In order to illustrate the effectiveness of the proposed method three individual industrial case studies are presented in this section. All necessary input data measurements were performed by means of a direct computer controlled CMM (*Mistral*, Brown & Sharpe-DEA) with ISO 10360-2 max. permissible error 3.5μm and PC-DMIS measurement software. A Renishaw PH10M head with TP200 probe and a 10mm length tip with diameter of 2mm were used. The number and distribution of sampling points conformed with the recommendations of BS7172:1989, (Flack**,** 2001), (9 points for planes and 15 for cylinders).

For a reverse engineered component of a working assembly (Part 2, Figure 7) assignment of dimensional tolerances was carried out using the developed methodology. The case study assembly of Figure 7 is incorporated in an optical sensor alignment system. Its' location, orientation and dynamic balance is considered of paramount importance for the proper function of the sensor. The critical assembly requirements that are here examined are the clearance gaps between the highlighted features (D1, D2, D3) of Part 1 – Shaft and Part 2- Hole in Figure 8. Four intact pairs of components were available for measurements. The analysis of section 3 was performed for all three critical features of Figure 8 individually. However, for the economy of the chapter, input data and method results are only presented for the D2 RE-feature, in Tables 2 and 3 respectively. The selected ISO 286 fits, Figure 9, produced in 12min *(10min CMM-measurements + 2min Computer aided implementation)* were experimentally verified and well approved by fitting reconstructed components in existing

provided by the machine shop DFF and the total accuracy cost is thus formulated as,

 1 1

tolerance chain constraints.

**5.2 Minimum machining cost** 

finally selected as the optimum one.

and in use assemblies.

**6. Application examples and case studies** 

**6.1 Application example of RE dimensional tolerancing** 

min max <sup>00</sup> *<sup>i</sup> DDftD <sup>j</sup>* (36)

/ min *<sup>i</sup>*

(37)

classification. In third level TEs are classified to external and internal ones as the achievement of internal tolerances usually results to higher accuracy costs. The fourth TE classification level distinguishes between plain and complex TEs depending on the absence or presence of additional feature details (grooves, wedges, ribs, threads etc). They do not change the principal TE geometry but they indirectly contribute to the increase of the accuracy cost. In the final fifth level, the involvement of the TE size to the accuracy cost is considered. TEs are classified, to the nominal size of the chain dimension, into six groups by integrating two sequential ISO 286-1 size ranges.

Fig. 6. Tolerance Elements five-class hierarch system (Dimitrellou et al., 2007a)

Based on the TE-method the actual machining accuracy capabilities and the relative cost per TE-class of a particular machine shop are recorded through an appropriately developed Database Feedback Form (DFF). The latter includes the accuracy cost for all the 96 TE-classes in the size range 3-500 mm and tolerances range IT6-IT10. DFF is filled once, at the system commissioning stage, by the expert engineers of the machine shop where the assembly components will be manufactured and can then be updated each time changes occur in the shop machines, facilities and/or expertise. The DFF data is then processed by the system through the least-square approximation and the system constructs and stores a costtolerance relationship of the power function type, per TE-class.

#### **5.1 Tolerance chain constrains**

In a n-member dimensional chain the tolerances of the individual dimensions D1,D2,…,Dn, control the variation of a critical end-dimension D0, according to the chain,

$$D\_0 = f(D\_1, D\_2, \dots, D\_n) \tag{34}$$

where f(D) can be either a linear or nonlinear function. To ensure that the end-dimension will be kept within its specified tolerance zone, the worst-case constrain that provides for 100% interchangeability has to be satisfied,

$$f\left(D\_{\,i\,\max} + D\_{\,j\,\min}\right) \le D\_0 + t\_0\tag{35}$$

$$D\_0 - t\_0 \le f\left(D\_{i\min} + D\_{j\max}\right) \tag{36}$$

where t0 is the tolerance of the end-dimension D0. For an (i+j)-member dimensional chain dimensions Di constitute the positive members of the chain while dimensions Dj constitute its negative members. In RE tolerancing, preferred alternatives for nominal sizes and dimensional tolerances that are generated from the analysis of Section 3, for each dimension involved in the chain are further filtered out by taking into consideration the above tolerance chain constraints.

#### **5.2 Minimum machining cost**

148 Reverse Engineering – Recent Advances and Applications

classification. In third level TEs are classified to external and internal ones as the achievement of internal tolerances usually results to higher accuracy costs. The fourth TE classification level distinguishes between plain and complex TEs depending on the absence or presence of additional feature details (grooves, wedges, ribs, threads etc). They do not change the principal TE geometry but they indirectly contribute to the increase of the accuracy cost. In the final fifth level, the involvement of the TE size to the accuracy cost is considered. TEs are classified, to the nominal size of the chain dimension, into six groups by

Fig. 6. Tolerance Elements five-class hierarch system (Dimitrellou et al., 2007a)

tolerance relationship of the power function type, per TE-class.

control the variation of a critical end-dimension D0, according to the chain,

**5.1 Tolerance chain constrains** 

100% interchangeability has to be satisfied,

Based on the TE-method the actual machining accuracy capabilities and the relative cost per TE-class of a particular machine shop are recorded through an appropriately developed Database Feedback Form (DFF). The latter includes the accuracy cost for all the 96 TE-classes in the size range 3-500 mm and tolerances range IT6-IT10. DFF is filled once, at the system commissioning stage, by the expert engineers of the machine shop where the assembly components will be manufactured and can then be updated each time changes occur in the shop machines, facilities and/or expertise. The DFF data is then processed by the system through the least-square approximation and the system constructs and stores a cost-

In a n-member dimensional chain the tolerances of the individual dimensions D1,D2,…,Dn,

where f(D) can be either a linear or nonlinear function. To ensure that the end-dimension will be kept within its specified tolerance zone, the worst-case constrain that provides for

0 12 ( , ,..., ) *D fD D D <sup>n</sup>* (34)

*f D D Dt i j* max min 0 0 (35)

integrating two sequential ISO 286-1 size ranges.

A second sorting out is applied by taking into account the accuracy cost for each combination of alternatives that obtained in the previous stage. Cost-tolerance functions are provided by the machine shop DFF and the total accuracy cost is thus formulated as,

$$\mathbf{C}\_{\text{total}} = \sum\_{i=1}^{n} \mathbf{C} \left( t\_i \right) = \sum\_{i=1}^{n} \left[ A\_i + B\_i \;/\; t\_i^{k\_i} \right] \to \min \tag{37}$$

where C(t) is the relative cost for the production of the machining tolerance ±t and A, B, k are constants. The combination of alternatives that corresponds to the minimum cost is finally selected as the optimum one.

#### **6. Application examples and case studies**

In order to illustrate the effectiveness of the proposed method three individual industrial case studies are presented in this section. All necessary input data measurements were performed by means of a direct computer controlled CMM (*Mistral*, Brown & Sharpe-DEA) with ISO 10360-2 max. permissible error 3.5μm and PC-DMIS measurement software. A Renishaw PH10M head with TP200 probe and a 10mm length tip with diameter of 2mm were used. The number and distribution of sampling points conformed with the recommendations of BS7172:1989, (Flack**,** 2001), (9 points for planes and 15 for cylinders).

#### **6.1 Application example of RE dimensional tolerancing**

For a reverse engineered component of a working assembly (Part 2, Figure 7) assignment of dimensional tolerances was carried out using the developed methodology. The case study assembly of Figure 7 is incorporated in an optical sensor alignment system. Its' location, orientation and dynamic balance is considered of paramount importance for the proper function of the sensor. The critical assembly requirements that are here examined are the clearance gaps between the highlighted features (D1, D2, D3) of Part 1 – Shaft and Part 2- Hole in Figure 8. Four intact pairs of components were available for measurements. The analysis of section 3 was performed for all three critical features of Figure 8 individually. However, for the economy of the chapter, input data and method results are only presented for the D2 RE-feature, in Tables 2 and 3 respectively. The selected ISO 286 fits, Figure 9, produced in 12min *(10min CMM-measurements + 2min Computer aided implementation)* were experimentally verified and well approved by fitting reconstructed components in existing and in use assemblies.

F10 / h10, F11 / h10, F12 / h10, G10 / h10, G11 / h10,

F10 / h11, F11 / h11, F12 / h11, G10 / h11, G11 / h11,

D10 / h9, D11 / h9, D10 / h8, D11 / h8, D10 / h11, D11

G12 / h10, H11 / h10, H12 / h10

G12 / h11, H11 / h11, H12 / h11

 *22.050* 

22.000

/ h11

Table 3. Sets of Suggested and Preferred fits for the case study

Fig. 9. Selected ISO 286 fits applied on the case study mechanical drawing

A new optical sensor made necessary the redesign of an existing bracket that had to be fastened on the old system through the original group of 4 x M5 threaded bolts, Figure 10. The accuracy of the location and orientation of the sensor bracket was considered critical for the sensor proper function. Bracket redesign had to be based on two available old and not used reference components. In the following, the method application is focused only on the allocation of position tolerances for the four bracket mounting holes. The problem represents, apparently, a typical fixed fastener case. For the chapter economy the input data and the produced results for the allocation of the position tolerance for the through hole H1, Figure 11, are only here presented and discussed. Standard diameter of the holes is 5.3mm ± 0.1mm and minimum clearance between hole and M5 screw 0.2mm.The weight coefficient

**6.2 Application example of RE geometrical tolerancing** 

in the relationships (21) was taken k=1.

Η11 / f11, Η11 / g7, Η11 / g8, Η11 / g9, Η11 / g10, Η11 / g11, Η11 / h8, Η11 /

Η11 / h10, Η11 / h11, Η12 / e9, Η12 /

Η12 / e11, Η12 / f10, Η12 / f11, Η12 /

Η12 / g8, Η12 / g9, Η12 / g10, Η12 / g11 Η12 / h8, Η12 / h9, Η12 / h10, Η12 / h11

Η11 / e9, Η11 / h8, Η11 / h9, Η11 / h11

D10 / h9, D11 / h9, D10 / h8, D11 / h8

h9,

e10,

g7,

**Preferred Fits**  22.050

22.000

Fig. 7. RE dimensional tolerancing case study parts and assembly

Fig. 8. Critical RE-features of the RE dimensional tolerancing case study parts


Table 2. Input data related to case study RE-features



Table 3. Sets of Suggested and Preferred fits for the case study

150 Reverse Engineering – Recent Advances and Applications

Fig. 7. RE dimensional tolerancing case study parts and assembly

 **D2**

Fig. 8. Critical RE-features of the RE dimensional tolerancing case study parts

 **D1**

**D3** 

**h #1** 22.136 0.008

**h #3** 22.091 0.003 **h #4** 22.078 0.004

**s #1** 21.998 0.003

**s #3** 21.979 0.006 **s #4** 21.972 0.005

> *22.000*

**Uh =** 0.009mm

**Us =** 0.012mm

Table 2. Input data related to case study RE-features

Η12 / g7, Η12 / g8, Η12 / g9, Η12 / g10, Η12 / g11, Η12 / h8, Η12 / h9, Η12 /

Η11 / e9, Η11 / e10, Η11 / e11, Η11 /

**Suggested Fits**  22.000

h10, Η12 / h11

f10,

22.050

**Hole** *dM\_h* **(mm)** *Fh* **(mm) Ra\_h** 

**D1**

**D2 D3** 

3.8 **h #2** 22.128 0.008

**Shaft** *dM\_s* **(mm)** *Fs* **(mm) Ra\_s** 

2.4 **s #2** 21.984 0.005

D10 / h8, D11 / h8, D12 / h8, E11/ h8, E12/ h8, F11 / h8,

D10 / h9, D11 / h9, D12 / h9, E11/ h8, E12/ h9, F11 / h9,

D10 / h10, D11 / h10, D12 / h10, E11/ h10, E12/ h10, F11 / h10, F12 / h10, G11 / h10, G12 / h10, H12 / h10 D10 / h11, D11 / h11, D12 / h11, E11/ h11, E12/ h11, F11 / h11, F12 / h11, G11 / h11, G12 / h11, H12 / h11

F12 / h8, G11 / h8, G12 / h8, H12 / h8

F12 / h9, G11 / h9, G12 / h9, H12 / h9

Fig. 9. Selected ISO 286 fits applied on the case study mechanical drawing

#### **6.2 Application example of RE geometrical tolerancing**

A new optical sensor made necessary the redesign of an existing bracket that had to be fastened on the old system through the original group of 4 x M5 threaded bolts, Figure 10. The accuracy of the location and orientation of the sensor bracket was considered critical for the sensor proper function. Bracket redesign had to be based on two available old and not used reference components. In the following, the method application is focused only on the allocation of position tolerances for the four bracket mounting holes. The problem represents, apparently, a typical fixed fastener case. For the chapter economy the input data and the produced results for the allocation of the position tolerance for the through hole H1, Figure 11, are only here presented and discussed. Standard diameter of the holes is 5.3mm ± 0.1mm and minimum clearance between hole and M5 screw 0.2mm.The weight coefficient in the relationships (21) was taken k=1.

 0.060 |Α|Β|C| 7.000 6.000 MMC 0.120 |Α|D|C| 30.950 50.950 MMC 0.060 |Α|C|B| 6.950 5.950 LMC 0.120 |Α|D|C| 30.900 51.000 LMC

 0.060 |Α|Β|D| 7.050 51.000 LMC 0.140 |Α|Β|C| 7.050 6.000 MMC 0.060 |Α|C|D| 30.950 6.000 MMC 0.140 |Α|C|B| 7.000 5.950 **-**  0.060 |Α|D|B| 31.000 51.000 MMC 0.140 |Α|C|B| 7.050 6.000 **-**  0.060 |Α|D|B| 31.050 51.050 LMC 0.140 |Α|Β|D| 7.000 51.100 LMC 0.060 |Α|D|C| 30.950 51.000 MMC 0.140 |Α|C|D| 31.050 6.000 MMC 0.060 |Α|D|C| 31.050 50.950 LMC 0.140 |Α|C|D| 31.100 6.000 LMC **. . . . . . . . . . .**  0.140 |Α|C|D| 31.100 6.050 LMC 0.080 |Α|Β|C| 7.000 6.000 - 0.140 |Α|D|B| 30.950 51.000 - 0.080 |Α|Β|C| 7.100 6.000 LMC 0.140 |Α|D|B| 30.900 51.000 MMC 0.080 |Α|C|B| 7.050 6.000 MMC 0.140 |Α|D|C| 30.950 50.950 - 0.080 |Α|Β|D| 7.000 51.000 **-**  0.140 |Α|D|C| 30.950 51.000 - 0.080 |Α|Β|D| 7.100 51.000 LMC 0.140 |Α|D|C| 30.900 51.000 MMC

 0.080 |Α|D|B| 31.000 51.000 MMC 0.160 |Α|Β|C| 6.950 5.950 - 0.080 |Α|D|B| 30.950 51.000 LMC 0.160 |Α|Β|C| 6.950 6.000 - 0.080 |Α|D|C| 31.000 6.000 - 0.160 |Α|C|B| 7.000 5.950 - 0.080 |Α|D|C| 30.095 51.050 MMC 0.160 |Α|C|B| 7.150 6.000 LMC 0.080 |Α|D|C| 31.050 51.050 LMC 0.160 |Α|Β|D| 7.000 51.100 MMC **. . . . . . . . . . .**  0.160 |Α|Β|D| 7.100 51.000 LMC 0.100 |Α|Β|C| 7.000 6.000 - 0.160 |Α|C|D| 30.950 6.000 **-**  0.100 |Α|C|B| 7.000 6.000 - 0.160 |Α|C|D| 30.900 5.900 LMC 0.100 |Α|C|B| 7.050 6.000 - 0.160 |Α|D|B| 31.000 51.050 **-**  0.100 |Α|Β|D| 7.000 51.000 - 0.160 |Α|D|C| 31.050 51.000 - 0.100 |Α|Β|D| 6.950 51.000 LMC 0.160 |Α|D|C| 30.900 51.000 MMC 0.100 |Α|C|D| 31.000 6.000 - 0.160 |Α|D|C| 30.950 50.900 LMC

 0.100 |Α|D|B| 31.000 51.050 MMC 0.180 |Α|Β|C| 7.050 6.000 **-**  0.100 |Α|D|B| 30.950 50.900 LMC 0.180 |Α|Β|C| 7.150 6.000 LMC 0.100 |Α|D|C| 30.950 51.000 LMC 0.180 |Α|C|B| 6.950 5.950 **-**  0.100 |Α|D|C| 31.050 50.950 LMC 0.180 |Α|C|B| 6.900 6.000 LMC **. . . . . . . . . . .**  0.180 |Α|Β|D| 6.900 51.000 MMC 0.120 |Α|Β|C| 7.050 6.050 MMC 0.180 |Α|Β|D| 6.900 51.050 MMC 0.120 |Α|Β|C| 7.100 6.000 LMC 0.180 |Α|C|D| 31.050 6.000 **-**  0.120 |Α|C|B| 7.050 5.900 LMC 0.180 |Α|C|D| 31.100 6.100 LMC 0.120 |Α|C|B| 7.100 5.950 LMC 0.180 |Α|D|B| 30.900 51.050 MMC 0.120 |Α|Β|D| 7.000 50.900 LMC 0.180 |Α|D|B| 31.050 51.000 MMC 0.120 |Α|C|D| 31.050 6.000 MMC 0.180 |Α|D|B| 31.100 51.000 LMC 0.120 |Α|C|D| 31.050 6.000 LMC 0.180 |Α|D|C| 30.950 51.000 **-**  0.120 |Α|D|B| 31.000 51.050 MMC 0.180 |Α|D|C| 31.050 50.950 MMC 0.120 |Α|D|B| 30.900 50.950 LMC 0.180 |Α|D|C| 30.900 50.900 LMC Table 5. Representative sample of application example suggested position tolerances

*Modifier TPOS DRF X Y Material* 

*Modifier* 

*TPOS DRF X Y Material* 

0.060 |Α|C|B| 7.100 6.000 LMC **. . . . . . . . . . .**

0.080 |Α|C|D| 31.000 6.000 - **. . . . . . . . . . .** 

0.100 |Α|C|D| 30.950 5.950 MMC **. . . . . . . . . . .** 

Fig. 10. Reference RE part and mating parts assembly

The CAD model of the redesigned new bracket was created in Solidworks taking into account the measured data of the reference components and, as well as, the new sensor mounting requirements. Datum and feature *(hole H1)* related input data are given in Table 4. Four candidate datum features A, B, C and D were considered, Figure 11. In step (a) 10 candidate DRFs (|Α|Β|C|, |Α|C|B|, |Α|Β|D|, |Α|D|B|, |Α|C|D|, |Α|D|C|, |D|A|C|, |D|A|B|, |D|B|A|, |D|C|A|) were produced by the algorithm for 7 position tolerance sizes, 0.06-0.18mm and consequently 10 sets of candidate theoretical dimensions, [*(CP(ij)X, CP(ij)Y)*, i=3,…9, j=1,…,10]. Negligible form and orientation deviations of datums A and D reduced the DRFs to the first six of them as |D|A|C|≡|Α|D|C|≡|D|C|A| and |D|A|B|≡|Α|D|B|≡|D|B|A|, having thus provided for 751 suggested tolerances in the following step (b). A representative sample of these tolerances is shown in Table 5. Although computational time difference is not significant, it is noticed that the quantity of the suggested results is strongly influenced by the number of the initially recognized possible datum features*,* weight coefficient *k* of the constraints (21), parameter *δ* of the equations (23) and the number of the available reference components.


Table 4. Input data related to case study feature and datum

152 Reverse Engineering – Recent Advances and Applications

The CAD model of the redesigned new bracket was created in Solidworks taking into account the measured data of the reference components and, as well as, the new sensor mounting requirements. Datum and feature *(hole H1)* related input data are given in Table 4. Four candidate datum features A, B, C and D were considered, Figure 11. In step (a) 10 candidate DRFs (|Α|Β|C|, |Α|C|B|, |Α|Β|D|, |Α|D|B|, |Α|C|D|, |Α|D|C|, |D|A|C|, |D|A|B|, |D|B|A|, |D|C|A|) were produced by the algorithm for 7 position tolerance sizes, 0.06-0.18mm and consequently 10 sets of candidate theoretical dimensions, [*(CP(ij)X, CP(ij)Y)*, i=3,…9, j=1,…,10]. Negligible form and orientation deviations of datums A and D reduced the DRFs to the first six of them as |D|A|C|≡|Α|D|C|≡|D|C|A| and |D|A|B|≡|Α|D|B|≡|D|B|A|, having thus provided for 751 suggested tolerances in the following step (b). A representative sample of these tolerances is shown in Table 5. Although computational time difference is not significant, it is noticed that the quantity of the suggested results is strongly influenced by the number of the initially recognized possible datum features*,* weight coefficient *k* of the constraints (21), parameter *δ* of the equations (23) and the

*dM (mm)* **Α|Β|C Α|C|B Α|Β|D Α|C|D Α|D|B Α|D|C** 

*Datum RFDF (mm) RO (mm) RODF (mm)*

*A* 0.008 0.011 0.005 0.006 B-0.024 /C-0.021 /D-

*B* 0.026 0.023 0.012 0.010 A-0.041 /C-0.039 /D-

*C* 0.016 0.021 0.016 0.011 A-0.042 /B-0.034 /D-

*D* 0.005 0.008 0.003 0.007 A-0.010 /B-0.022 /C-

Table 4. Input data related to case study feature and datum

XM1 7.023 7.031 7.025 31.014 31.007 31.011 YM1 5.972 5.961 50.981 5.964 50.980 50.978

XM2 6.988 7.004 6.987 30.973 30.962 30.971 YM2 6.036 6.019 51.028 6.017 51.026 51.012

*Part 1 Part 2 Part 1 Part 2 Part 1 Part 2* 

0.008

0.035

0.041

0.020

B-0.027/C-0.019 /D-0.007

A-0.046/C-0.043 /D-0.042

A-0.038/B-0.036 /D-0.033

A-0.009/ B-0.025 /C-0.017

Fig. 10. Reference RE part and mating parts assembly

number of the available reference components.

*Part1* 5.291

*Part2* 5.244


Table 5. Representative sample of application example suggested position tolerances

component design and manufacturing information is, however, not available and the dimensional accuracy specifications for component B reconstruction have to be

The machine shop where the part will be manufactured has an IT6 best capability and its DFF processed and the results stored. Alternatives for parts A, C and B, provided by the RE dimensional analysis of Section 3, are shown in Table 7(a) and (b) respectively. The 64 possible combinations of the part B alternatives are filtered out according to the tolerance chain constrains and, as a result, 24 combinations occur for the dimensions and tolerances D5±t5, D6±t6, D7±t7 as shown in Table 8. The optimum combination that corresponds to the

reestablished.

minimum accuracy cost is the combination 64.

Fig. 12. Application example of Cost-Effective RE tolerancing

Table 7. Dimensional alternatives for parts A, C and B

*Alternative 1 Alternative 2 Dmin Dmax Dmin Dmax* D1 190.01 190.04 189.97 190.00 D2 14.99 15.02 14.98 15.00 D3 45.01 45.02 44.98 45.00 D7 20.00 20.01 19.99 20.005 D10 12.01 12.025 11.98 12.00 D11 97.00 97.02 96.98 97.025

*a/a D5 D6 D9*

*Dmin Dmax Dmin Dmax Dmin Dmax* 1 14.00 14.12 95.03 95.15 74.67 75.23 2 13.89 14.30 94.98 95.11 75.00 75.01 3 13.98 13.99 94.86 94.99 74.77 74.99 4 13.89 14.01 94.95 95.14 74.98 75.10


Table 6. Set of preferred position tolerances for the application example

The preferred 11 out of 751 position tolerances of the Table 6 were obtained applying the selection options and guidelines of the section 4.4. Parallel results were obtained for the other three holes. As it came out, all of them belong to a group of holes with common DRF. The position tolerance size 0.100mm and the DRF |Α|Β|C|were finally chosen by the machine shop. Theoretical hole location dimensions are shown in Figure 11. The results were experimentally verified and approved. Time needed for the entire task was 12min (CMM) + 6min (Analysis) =18min. The usual trial-and-error way would, apparently, require considerably longer time and produce doubtful results. Reliability of the results can certainly be affected by failing to recognize initial datum features. In machine shop practice however, risk for something like that is essentially negligible.

Fig. 11. Case study datums (a) and the selected position tolerance of the case study (b)

#### **6.3 Application example of cost-effective RE tolerancing**

In the assembly of components A-B-C of Figure 12 the dimension D4 = 74.95 ± 0.25mm is controlled through the dimensional chain,

$$D\_4 = \sin D\_3 \left\{ D\_1 + D\_2 + D\_7 + D\_{10} \cdot D\_{11} \cdot D\_5 \cdot D\_6 + D\_9 \right\}$$

with D1 = 190mm, D2 = 15mm, D3 = 45°, D5 = 14mm, D6 = 95mm, D7 = 20mm, D9 = 75mm, D10 = 12mm, D11 = 97mm. Component B is reverse engineered and needs to be remanufactured with main intention to fit and perform well in the existing assembly. All of the original 154 Reverse Engineering – Recent Advances and Applications

The preferred 11 out of 751 position tolerances of the Table 6 were obtained applying the selection options and guidelines of the section 4.4. Parallel results were obtained for the other three holes. As it came out, all of them belong to a group of holes with common DRF. The position tolerance size 0.100mm and the DRF |Α|Β|C|were finally chosen by the machine shop. Theoretical hole location dimensions are shown in Figure 11. The results were experimentally verified and approved. Time needed for the entire task was 12min (CMM) + 6min (Analysis) =18min. The usual trial-and-error way would, apparently, require considerably longer time and produce doubtful results. Reliability of the results can certainly be affected by failing to recognize initial datum features. In machine shop practice

 0.080 |Α|Β|C| 7.000 6.000 - 0.100 |Α|C|B| 7.000 6.000 - 0.800 |Α|Β|D| 7.000 51.000 - 0.100 |Α|C|B| 7.050 6.000 - 0.080 |Α|C|D| 31.000 6.000 - 0.100 |Α|Β|D| 7.000 51.000 **-**  0.080 |Α|D|C| 31.000 51.000 - 0.100 |Α|C|D| 31.000 6.000 **-**  0.100 |Α|Β|C| 7.000 6.000 - 0.100 |Α|D|B| 31.000 51.000 **-**  0.100 |Α|D|C| 31.000 51.000 **-** 

Table 6. Set of preferred position tolerances for the application example

however, risk for something like that is essentially negligible.

(a) (b)

**6.3 Application example of cost-effective RE tolerancing** 

controlled through the dimensional chain,

Fig. 11. Case study datums (a) and the selected position tolerance of the case study (b)

( ) <sup>4</sup> +++sin= <sup>+</sup> *DDDDDDDDDD* <sup>96511107213</sup> ---

In the assembly of components A-B-C of Figure 12 the dimension D4 = 74.95 ± 0.25mm is

with D1 = 190mm, D2 = 15mm, D3 = 45°, D5 = 14mm, D6 = 95mm, D7 = 20mm, D9 = 75mm, D10 = 12mm, D11 = 97mm. Component B is reverse engineered and needs to be remanufactured with main intention to fit and perform well in the existing assembly. All of the original

*Modifier TPOS DRF X Y Material* 

*Modifier* 

*TPOS DRF X Y Material* 

component design and manufacturing information is, however, not available and the dimensional accuracy specifications for component B reconstruction have to be reestablished.

The machine shop where the part will be manufactured has an IT6 best capability and its DFF processed and the results stored. Alternatives for parts A, C and B, provided by the RE dimensional analysis of Section 3, are shown in Table 7(a) and (b) respectively. The 64 possible combinations of the part B alternatives are filtered out according to the tolerance chain constrains and, as a result, 24 combinations occur for the dimensions and tolerances D5±t5, D6±t6, D7±t7 as shown in Table 8. The optimum combination that corresponds to the minimum accuracy cost is the combination 64.

Fig. 12. Application example of Cost-Effective RE tolerancing


Table 7. Dimensional alternatives for parts A, C and B

In industrial manufacturing, tolerance assignment is one of the key activities in the product creation process. However, tolerancing is much more difficult to be successfully handled in RE. In this case all or almost all of the original component design and manufacturing information is not available and the dimensional and geometric accuracy specifications for component reconstruction have to be re-established, one way or the other, practically from scratch. RE-tolerancing includes a wide range of frequently met industrial manufacturing problems and is a task that requires increased effort, cost and time, whereas the results, usually obtained by trial-and-error, may well be not the best. The proposed methodology offers a systematic solution for this problem in reasonable computing time and provides realistic and industry approved results. This research work further extends the published research on this area by focusing on type of tolerances that are widely used in industry and almost always present in reverse engineering applications. The approach, to the extent of the author's knowledge, is the first of the kind for this type of RE problems that can be directly implemented within a CAD environment. It can also be considered as a pilot for further research and development in the area of RE tolerancing. Future work is oriented towards the computational implementation of the methodology in 3D-CAD environment, the RE *composite* position tolerancing that concerns patterns of repetitive features, the methodology application on the whole range of GD&T types and the integration of function oriented wear simulation models in order to evaluate input data that come from RE parts that bear

Abella, R.J.; Dashbach, J.M. &, McNichols, R.J. (1994). Reverse Engineering Industrial

Anselmetti, B. & Louati, H. (2005). Generation of manufacturing tolerancing with ISO

ASME Standard. (1994). *Y14.5.1M-1994: Mathematical Definition of Dimensioning and* 

ASME Standard. (2009). *Y14.5M–2009: Dimensioning and Tolerancing*, The American Society

Bagci, E. (2009). Reverse engineering applications for recovery of broken or worn parts and

Borja, V., Harding, J.A. & Bell, B. (2001). A conceptual view on data-model driven reverse

Chant, A.; Wilcock, D. & Costello, D. (1998). The Determination of IC engine inlet port

Applications, *Computers & Industrial Engineering*, Vol.26, No. 2, pp. 381-385, ISSN:

standards, *International Journal of Machine Tools & Manufacture,* Vol. 45, pp. 1124–

*Tolerancing Principles*, The American Society of Mechanical Engineers, New York,

re-manufacturing: Three case studies, *Advances in Engineering Software*, Vol. 40, pp.

geometries by Reverse Engineering, *International Journal of Advanced Manufacturing* 

**7. Conclusion** 

considerable amount of wear.

0360-8352

USA

1131, ISSN: 0890-6955

407–418, ISSN: 0965-9978

of Mechanical Engineers, New York, USA

engineering, *Int. J. Prod. Res.,* Vol.39, No 4, pp. 667-687

*Technology,* Vol.14, pp. 65–69, ISSN: 0268-3768

**8. References** 


Table 8. Filtered out combinations
