**2. Literature review**

The purpose and the main application areas of RE along with current methodologies and practical solutions for reverse engineering problems in industrial manufacturing are identified and discussed in several reference publications, e.g (Ingle, 1994; Raja & Fernandes, 2008; Wego, 2011). Moreover, the application of RE techniques and their implementation on modern industrial engineering practice is the subject of a numerous research works, e.g. (Abella et al., 1994; Bagci, 2009; Dan & Lancheng, 2006; Endo, 2005; Zhang, 2003). In that context, RE methodologies are applied for the reconstruction of 134 Reverse Engineering – Recent Advances and Applications

tolerance design is well understood in the engineering community it still remains an engineering task that largely depends on experimental data, industrial databases and guidelines, past experience and individual expertise, (Kaisarlis et al., 2008). Geometrical and dimensional tolerances are of particular importance, on the other hand, not only in industrial production but also in product development, equipment upgrading and maintenance. The last three activities include, inevitably, RE tasks which go along with the reconstruction of an object CAD model from measured data and have to do with the assignment of dimensional and geometrical manufacturing tolerances to this object. In that context, tolerancing of RE components address a wide range of industrial applications and real-world manufacturing problems such as tolerance allocation in terms of the actual functionality of a prototype assembly, mapping of component experimental design modifications, spare part tolerancing for machines that are out of production or need improvements and no drawings are available, damage repair, engineering maintenance etc. The objective of remanufacturing a needed mechanical component which has to fit and well perform in an existing assembly and, moreover, has to observe the originally assigned functional characteristics of the product is rather delicate. The objective in such applications is the designation of geometric and dimensional tolerances that match, as closely as possible, to the original *(yet unknown)* dimensional and geometrical accuracy specifications that reveal the original design intend. RE tolerancing becomes even more sophisticated in case that Coordinate Measuring Machines' (CMM) data of a few or just only one of the original components to be reversibly engineered are within reach. Moreover, if operational use has led to considerable wear/ damage or one of the mating parts is missing, then the complexity of the problem increases considerably. The RE tolerancing problem has not been sufficiently and systematically addressed to this date. Currently, in such industrial problems where typically relevant engineering information does not exist, the conventional trial and error approach for the allocation of RE tolerances is applied. This approach apparently requires much effort and time and offers no guarantee for the generation of the best of results.

This research work provides a novel, modern and integrated methodology for tolerancing in RE. The problem is addressed in a systematic, time and cost efficient way, compatible with the current industrial practice. The rest of the chapter is organized as follows: after the review of relevant technical literature in Section 2, the theoretical analysis of RE dimensional and geometrical tolerancing is presented *(Sections 3 and 4 respectively)*. The application of Tolerance Elements (TE) method for cost-effective, competent tolerance designation in RE is then introduced in Section 5. Certain application examples that illustrate the effectiveness of the methodology are further presented and discussed in Section 6. Main conclusions and

The purpose and the main application areas of RE along with current methodologies and practical solutions for reverse engineering problems in industrial manufacturing are identified and discussed in several reference publications, e.g (Ingle, 1994; Raja & Fernandes, 2008; Wego, 2011). Moreover, the application of RE techniques and their implementation on modern industrial engineering practice is the subject of a numerous research works, e.g. (Abella et al., 1994; Bagci, 2009; Dan & Lancheng, 2006; Endo, 2005; Zhang, 2003). In that context, RE methodologies are applied for the reconstruction of

future work orientation are included in the final Section 7 of the chapter.

**2. Literature review** 

mechanical components and assemblies that have been inevitably modified during several stages of their life cycle, e.g. surface modifications of automotive components during prototype functionality testing (Chant et al., 1998; Yau, 1997), mapping of sheet metal forming deviations on free form surface parts (Yuan et al., 2001), monitoring on the geometrical stability during test runs of mock-up turbine blades used in nuclear power generators (Chen & Lin, 2000), repair time compression by efficient RE modeling of the broken area and subsequent rapid spare part manufacturing (Zheng et al., 2004), recording of die distortions due to thermal effects for the design optimization of fan blades used in aero engines (Mavromihales et al., 2003).

The principles and applications of tolerancing in modern industrial engineering can also be found in several reference publications, e.g. (Drake, 1999; Fischer, 2004). An extensive and systematic review of the conducted research and the state of the art in the field of dimensioning and tolerancing techniques is provided by several recent review papers, e.g. (Singh et al. 2009a, 2009b) and need not be reiterated here. In the large number of research articles on various tolerancing issues in design for manufacturing that have been published over the last years, the designation of geometrical tolerances has been adequately studied under various aspects including tolerance analysis and synthesis, composite positional tolerancing, geometric tolerance propagation, datum establishment, virtual manufacturing, inspection and verification methods for GD&T specifications, e.g. (Anselmetti and Louati, 2005; Diplaris & Sfantsikopoulos, 2006; Martin et al., 2011).

Although RE-tolerancing is a very important and frequently met industrial problem, the need for the development of a systematic approach to extract appropriate design specifications that concern the geometric accuracy of a reconstructed component has been only recently pointed out, (Borja et al., 2001; Thompson et al., 1999; VREPI, 2003). In one of the earliest research works that systematically addresses the RE-tolerancing problem (Kaisarlis et al., 2000), presents the preliminary concept of a knowledge-based system that aims to the allocation of standard tolerances as per ISO-286. The issue of datum identification in RE geometric tolerancing is approached in a systematic way by (Kaisarlis et al, 2004) in a later publication. Recently, (Kaisarlis et al, 2007, 2008) have further extend the research on this area by focusing on the RE assignment of position tolerances in the case of fixed and floating fasteners respectively. The methodology that is presented in this Chapter further develops the approach that is proposed on these last two publications. The novel contribution reported here deals with *(i)* the systematic assignment of *both* geometrical and dimensional tolerances in RE and their possible interrelation through the application of material modifiers on *both* the RE features and datums and *(ii)* the consideration of costeffective, competent tolerance designation in RE in a systematic way.

#### **3. Dimensional tolerancing in reverse engineering**

Reconstructed components must obviously mate with the other components of the mechanical assembly that they belong to, in *(at least)* the same way as their originals, in order the original assembly clearances to be observed, i.e. they must have appropriate manufacturing tolerances. As pointed out in Section 1, this is quite different and much more difficult to be achieved in RE than when designing from scratch where, through normal tolerance analysis/synthesis techniques and given clearances, critical tolerances are assigned, right from the beginning, to all the assembly components. Integration of geometric

Δh\_max = max{(max*RdM\_h* – min*RdM\_h*), max*RFh*, (60*·*mean*RRah*), *Uh*},

Δs\_max = max{(max*RdM\_s* - min*RdM\_s*), max*RFs*, (60*·*mean*RRas*), *Us*} In this step, the analysis is directed on the assignment of either ISO 286-1 clearance fits or

where |a| is the absolute value of the maximum ISO 286 clearance Fundamental Deviation (FD) for the relevant nominal sizes range *(the latter is approximated by the mean value of RdM\_h, RdM\_s sets)*. If the above condition is *not* satisfied the analysis is exclusively directed on ISO 2768 general tolerances. Otherwise, the following two cases are distinguished, *(i)* Δmax ≤ IT 11 and *(ii)* IT 11< Δmax ≤ IT 18. In the first case the analysis aims only on ISO 286 fits, whereas

The starting point for the *Step (b)* of the analysis is the production of the Candidate tolerance grades sets, *ITCAN\_h*, *ITCAN\_s*, for the hole and shaft features respectively. It is achieved by filtering the *initial Candidate IT grades* set, *ITCAN\_INIT*, which includes all standardized IT grades from IT01 to IT18, by the following conditions *(applied for both the h and s indexes),* 

Moreover, in case when estimated maximum and minimum functional clearance limits are

The above constraints are applied separately for the hole and shaft and qualify the members of the *ITCAN\_h*, *ITCAN\_s* sets. Likewise, the set of *initial* Candidate Fundamental Deviations, *FDCAN\_INIT*, that contains all the FDs applicable to clearance fits i.e. *FDCAN\_INIT* = {a, b, c, cd, d,

available (maxCL, minCL), candidate IT grades are qualified by the validation of,

e, f, fg, g, h}, is filtered by the constraints,

obtained from the integral part of the following equations,

ITCAN *<* maxCL

 FDCAN ≥ minCL

NSCAN\_h\_1 = int [ min*RdM\_h* – max FDCAN - maxITCAN\_h]

The latter constraints, (5), apparently only apply in case of maxCL and/or minCL availability. All qualified FDs are included in the common set of Candidate Fundamental Deviations, *FDCAN*. In the final stage of this step, the Candidate Nominal Sizes Sets, *NSCAN\_h*, *NSCAN\_s*, are initially formulated for the *hole* and *shaft* respectively. Their first members are

Δh\_max + Δs\_max + |a| ≥ max*RdM\_h* – min*RdM\_s* (1)

ITCAN <sup>≥</sup> max*RF* , ITCAN <sup>≥</sup> max*RdM* - min*Rd<sup>M</sup>* (2) ITCAN <sup>≤</sup> 60·mean*RRa* , ITCAN <sup>≥</sup> *<sup>U</sup>*

ITCAN *<sup>&</sup>lt;*maxCL – minCL (3)

FDCAN ≤ min*RdM\_h* – max*RdM\_s* (4)

FDCAN < maxCL – (min *ITCAN\_h* + min*ITCAN\_s*) (5)

NSCAN\_s\_1 = int [ max*RdM\_s* + max FDCAN + maxITCAN\_s] (6)

ISO 2768 general tolerances through the validation of the condition,

in the second case, *both* ISO 286 and ISO 2768 RE tolerances are pursued.

**3.2 Sets of candidate IT grades, fundamental deviations and nominal sizes** 

accuracy constrains aimed at the reconstruction of 3D models of RE-conventional engineering objects from range data has been studied adequately, (Raja & Fernandes, 2008; Várady et al., 1997). These studies deal, however, with the mathematical accuracy of the reconstructed CAD model by fitting curves and surfaces to 3D measured data. Feature– based RE (Thompson et al., 1999; VREPI, 2003) does not address, on the other hand, until now issues related with the manufacturing tolerances which have to be assigned on the CAD drawings in order the particular object to be possible to be made as required. A methodology for the problem treatment is proposed in the following sections.

Engineering objects are here classified according to their shape either as *free-form objects* or as *conventional engineering objects* that typically have simple geometric surfaces *(planes, cylinders, cones, spheres and tori)* which meet in sharp edges or smooth blends. In the following, Feature–Based RE for mechanical assembly components of the latter category is mainly considered. Among features of size (ASME, 2009), *cylindrical features* such as holes in conjunction with pegs, pins or (screw) shafts are the most frequently used for critical functions as are the alignment of mating surfaces or the fastening of mating components in a mechanical assembly. As a result, their role is fundamental in mechanical engineering and, consequently, they should be assigned with appropriate dimensional and geometrical tolerances. In addition, the stochastic nature of the manufacturing deviations makes crucial, for the final RE outcome, the quantity of the available (same) components that serve as reference for the measurements. The more of them are available the more reliable will be the results. For the majority of the RE cases, however, their number is extremely limited and usually ranges from less than ten to only one available item. Mating parts can also be inaccessible for measurements and there is usually an apparent lack of adequate original design and/or manufacturing information. In the scope of this research work, the developed algorithms address the full range of possible scenarios, from "only one original component – no mating component available" to "two or more original pairs of components available", focusing on parts for which either an ISO 286-1 clearance fit (of either hole or shaft basis system) or ISO 2768 (general tolerances) were originally designated.

Assignment of RE dimensional tolerances is accomplished by the present method in five sequential steps. In the primary step (a) the analysis is appropriately directed to ISO fits or general tolerances. In the following steps, the candidate *(Step b)*, suggested *(Step c)* and preferred *(Step d)* sets of RE-tolerances are produced. For the final RE tolerance selection *(Step e)* the cost-effective tolerancing approach, introduced in Section 5, is taken into consideration. For the economy of the chapter, the analysis is only presented for the "*two or more original pairs of components available"* case, focused on ISO 286 fits, as it is considered the most representative.

#### **3.1 Direction of the analysis on ISO fits and/or general tolerances**

Let *RdM\_h*, *RFh*, *RRah*, *Uh* and *RdM\_s*, *RFs*, *RRas*, *Us* be the sets of the measured diameters, form deviations, surface roughness, and the uncertainty of CMM measurements for the RE*hole* and the RE-*shaft* features respectively. The *Δmax, Δh\_max, Δs\_max* limits are calculated by,

Δmax= max{(max*RdM\_h* – min*RdM\_h*), (max*RdM\_s* - min*RdM\_s*), max*RFh*, max*RFs*, (60*·*mean*RRah*), (60·mean*RRas*), *Uh ,Us*},

136 Reverse Engineering – Recent Advances and Applications

accuracy constrains aimed at the reconstruction of 3D models of RE-conventional engineering objects from range data has been studied adequately, (Raja & Fernandes, 2008; Várady et al., 1997). These studies deal, however, with the mathematical accuracy of the reconstructed CAD model by fitting curves and surfaces to 3D measured data. Feature– based RE (Thompson et al., 1999; VREPI, 2003) does not address, on the other hand, until now issues related with the manufacturing tolerances which have to be assigned on the CAD drawings in order the particular object to be possible to be made as required. A

Engineering objects are here classified according to their shape either as *free-form objects* or as *conventional engineering objects* that typically have simple geometric surfaces *(planes, cylinders, cones, spheres and tori)* which meet in sharp edges or smooth blends. In the following, Feature–Based RE for mechanical assembly components of the latter category is mainly considered. Among features of size (ASME, 2009), *cylindrical features* such as holes in conjunction with pegs, pins or (screw) shafts are the most frequently used for critical functions as are the alignment of mating surfaces or the fastening of mating components in a mechanical assembly. As a result, their role is fundamental in mechanical engineering and, consequently, they should be assigned with appropriate dimensional and geometrical tolerances. In addition, the stochastic nature of the manufacturing deviations makes crucial, for the final RE outcome, the quantity of the available (same) components that serve as reference for the measurements. The more of them are available the more reliable will be the results. For the majority of the RE cases, however, their number is extremely limited and usually ranges from less than ten to only one available item. Mating parts can also be inaccessible for measurements and there is usually an apparent lack of adequate original design and/or manufacturing information. In the scope of this research work, the developed algorithms address the full range of possible scenarios, from "only one original component – no mating component available" to "two or more original pairs of components available", focusing on parts for which either an ISO 286-1 clearance fit (of either hole or shaft basis

Assignment of RE dimensional tolerances is accomplished by the present method in five sequential steps. In the primary step (a) the analysis is appropriately directed to ISO fits or general tolerances. In the following steps, the candidate *(Step b)*, suggested *(Step c)* and preferred *(Step d)* sets of RE-tolerances are produced. For the final RE tolerance selection *(Step e)* the cost-effective tolerancing approach, introduced in Section 5, is taken into consideration. For the economy of the chapter, the analysis is only presented for the "*two or more original pairs of components available"* case, focused on ISO 286 fits, as it is considered the

Let *RdM\_h*, *RFh*, *RRah*, *Uh* and *RdM\_s*, *RFs*, *RRas*, *Us* be the sets of the measured diameters, form deviations, surface roughness, and the uncertainty of CMM measurements for the RE*hole* and the RE-*shaft* features respectively. The *Δmax, Δh\_max, Δs\_max* limits are calculated by,

Δmax= max{(max*RdM\_h* – min*RdM\_h*), (max*RdM\_s* - min*RdM\_s*), max*RFh*, max*RFs*, (60*·*mean*RRah*), (60·mean*RRas*), *Uh ,Us*},

methodology for the problem treatment is proposed in the following sections.

system) or ISO 2768 (general tolerances) were originally designated.

**3.1 Direction of the analysis on ISO fits and/or general tolerances** 

most representative.

Δh\_max = max{(max*RdM\_h* – min*RdM\_h*), max*RFh*, (60*·*mean*RRah*), *Uh*},

$$\Delta\_{\mathsf{e}\\_\max} = \max\{ (\mathsf{max} \mathsf{R} d\_{\mathsf{M}\\_\text{s}} \cdot \mathsf{min} \mathsf{R} d\_{\mathsf{M}\\_\text{s}}), \max \mathsf{R} F\_{\text{s}} \mid (60 \cdot \mathsf{mean} \mathsf{R} \mathsf{R} a\_{\text{s}}), \mathsf{U}\_{\text{s}} \}$$

In this step, the analysis is directed on the assignment of either ISO 286-1 clearance fits or ISO 2768 general tolerances through the validation of the condition,

$$
\Delta\_{\text{h\\_max}} + \Delta\_{\text{s\\_max}} + \lfloor \text{a} \rfloor \ge \max \mathbf{R} d\_{\text{M\\_h}} - \min \mathbf{R} d\_{\text{M\\_s}} \tag{1}
$$

where |a| is the absolute value of the maximum ISO 286 clearance Fundamental Deviation (FD) for the relevant nominal sizes range *(the latter is approximated by the mean value of RdM\_h, RdM\_s sets)*. If the above condition is *not* satisfied the analysis is exclusively directed on ISO 2768 general tolerances. Otherwise, the following two cases are distinguished, *(i)* Δmax ≤ IT 11 and *(ii)* IT 11< Δmax ≤ IT 18. In the first case the analysis aims only on ISO 286 fits, whereas in the second case, *both* ISO 286 and ISO 2768 RE tolerances are pursued.

#### **3.2 Sets of candidate IT grades, fundamental deviations and nominal sizes**

The starting point for the *Step (b)* of the analysis is the production of the Candidate tolerance grades sets, *ITCAN\_h*, *ITCAN\_s*, for the hole and shaft features respectively. It is achieved by filtering the *initial Candidate IT grades* set, *ITCAN\_INIT*, which includes all standardized IT grades from IT01 to IT18, by the following conditions *(applied for both the h and s indexes),* 

$$\begin{aligned} \text{IT}\_{\text{CAN}} & \succeq \text{max} \mathbf{RF}\_{\prime} & \text{IT}\_{\text{CAN}} \ge \text{max} \mathbf{R} \mathbf{d}\_{M} \text{--} \min \mathbf{R} \mathbf{d}\_{M} \\ \text{IT}\_{\text{CAN}} & \in \mkern-1.12 \mathbf{f} \text{\$\! \$\mathbf{0}\$-mean} \mathbf{R} \mathbf{R} \mathbf{a}\_{/} \text{--} & \text{IT}\_{\text{CAN}} \ge \mathbf{U} \end{aligned} \tag{2}$$

Moreover, in case when estimated maximum and minimum functional clearance limits are available (maxCL, minCL), candidate IT grades are qualified by the validation of,

$$\begin{array}{c} \text{IT}\_{\text{CAN}} \mathsf{<}\text{maxCL} \\ \text{IT}\_{\text{CAN}} \mathsf{<}\text{maxCL} \text{ -}\text{minCL} \end{array} \tag{3}$$

The above constraints are applied separately for the hole and shaft and qualify the members of the *ITCAN\_h*, *ITCAN\_s* sets. Likewise, the set of *initial* Candidate Fundamental Deviations, *FDCAN\_INIT*, that contains all the FDs applicable to clearance fits i.e. *FDCAN\_INIT* = {a, b, c, cd, d, e, f, fg, g, h}, is filtered by the constraints,

$$\text{FD}\_{\text{CAN}} \le \min \text{Rd}\_{\text{M}\_{\text{J}}h} - \max \text{Rd}\_{\text{M}\_{\text{J}}s} \tag{4}$$

$$\begin{aligned} \text{FD}\_{\text{CAN}} & \succeq \text{minCL} \\ \text{FD}\_{\text{CAN}} & \leqslant \text{maxCL} \text{ - (min } IT\_{\text{CAN},h} + \text{minLT}\_{\text{CAN},s}) \end{aligned} \tag{5}$$

The latter constraints, (5), apparently only apply in case of maxCL and/or minCL availability. All qualified FDs are included in the common set of Candidate Fundamental Deviations, *FDCAN*. In the final stage of this step, the Candidate Nominal Sizes Sets, *NSCAN\_h*, *NSCAN\_s*, are initially formulated for the *hole* and *shaft* respectively. Their first members are obtained from the integral part of the following equations,

$$\begin{array}{l} \text{NS\_{CAN\\_h.1}} = \text{int} \left[ \min \mathbf{R} d\_{\text{M\\_h}} - \max \text{FD\_{CAN\\_}} \cdot \max \text{IT}\_{\text{CAN\\_h}} \right] \\ \text{NS\_{CAN\\_s.1}} = \text{int} \left[ \max \mathbf{R} d\_{\text{M\\_s}} + \max \text{FD\_{CAN\\_}} + \max \text{IT}\_{\text{CAN\\_s}} \right] \end{array} \tag{6}$$

min*RdM\_h*  NSCAN\_n+ FDCAN\_q ∀ FDCAN\_q ∈ *FD*CAN (14)

NSCAN\_n + FDCAN\_q + ITCAN\_h\_<sup>κ</sup> max*RdM\_h* ∀ FDCAN\_q ∈ *FD*CAN (15)

A limited number of *Preferred Fits* out of the *Suggested* ones is proposed in Step (d) through the consideration of ISO proposed fits. Moreover, the implementation of manufacturing guidelines, such as the fact that it is useful to allocate a slightly larger tolerance to the hole than the shaft, preference of Basic Hole fits over Basic Shaft ones, preference of nominal sizes that are expressed in integers or with minimum possible decimal places etc, are additionally used to "filter" the final range of the preferred fits. The final selection, Step (e), out of the limited set of preferred fits and the method end result is reached by the consideration of the machine shop capabilities and expertise in conjunction with the application of the cost – effective RE

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

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

(a) (b)

tolerancing approach, presented in Section 5 of the chapter.

**4. Geometrical tolerancing in reverse engineering** 

coaxiality or symmetry and, as well as, for run-out or profile tolerances.

Fig. 1. Suggested Basic Hole fits qualification

**3.4 Sets of preferred fits** 

ζ=1, 2, …, j 1 ≤ j ≤ 20

κ=1, 2, …, i 1≤ i ≤ 20

Following members of the sets are then calculated by an incremental increase, *δ*, of NSCAN\_h\_1 and NSCAN\_s\_1,

$$\begin{array}{c} \text{NS\_{CAN},h.,2} = \text{NS\_{CAN},h.1} + \delta \\ \text{NS\_{CAN},h.3} = \text{NS\_{CAN},h.2} + \delta \\ \text{I.3.2} = \text{NS\_{CAN},h.3} + \delta \\ \text{NS\_{CAN},h.y.} = \text{NS\_{CAN},h.v.1} + \delta \end{array} \qquad \begin{array}{c} \text{NS\_{CAN},s.2} = \text{NS\_{CAN},s.1} - \delta \\ \text{NS\_{CAN},s.3} = \text{NS\_{CAN},s.2} - \delta \\ \text{I.3.2} = \text{NS\_{CAN},s.\mu} - \delta \\ \text{NS\_{CAN},s.\mu} = \text{NS\_{CAN},s.\mu} - \delta \end{array} \tag{7}$$

bounded by,

$$\begin{aligned} \text{NS}\_{\text{CAN},h\_{-}\text{v}} & \leq \min \mathbf{R} d\_{\text{M},h} \\ \text{NS}\_{\text{CAN},s,\mu} & \geq \max \mathbf{R} d\_{\text{M},s} \end{aligned} \tag{8}$$

with the populations *ν, μ* not necessarily equal. In the relevant application example of section 6, *δ* is taken *δ=0.05mm*. Other *δ-*values can be, obviously, used depending on the case. Since both hole and shaft have a common nominal size in ISO-286 fits, the Candidate Nominal Sizes Set, *NSCAN*, is then produced by the common members of *NSCAN\_h*, *NSCAN\_s*,

$$\text{NS}\_{\text{CAN}} = \text{NS}\_{\text{CAN}\_{\text{-}}h} \cap \text{NS}\_{\text{CAN}\_{-}s} \tag{9}$$

#### **3.3 Sets of suggested fits**

In Step (c) of the analysis, a combined qualification for the members of the *IT*CAN\_h, *IT*CAN\_s, *FDCAN* and *NS*CAN sets is performed in order to produce the two sets of *suggested* Basic Hole, *BH*SG, and Basic Shaft *BS*SG fits. The members of *IT*CAN\_h and *IT*CAN\_s sets are sorted in ascending order. For the production of the *BH*SG set, every candidate nominal size of the *NS*CAN set is initially validated against all members of the *IT*CAN\_h set, Figure 1(a),

$$\text{NS}\_{\text{CAN},\text{n}} + \text{IT}\_{\text{CAN},\text{h}\_{\text{n}}} \ge \text{max} \mathbf{R} d\_{\text{M}\_{\text{h}}} \quad \forall \text{ NS}\_{\text{CAN},\text{n}} \in \text{NS}\_{\text{CAN}} \tag{10}$$

κ=1, 2, …, i 1≤ i ≤ 20

In case no member of the *IT*CAN\_h set satisfies the condition (10) for a particular NSCAN\_n, the latter is excluded from the *BH*SG set. In order to qualify for the *BH*SG set, candidate nominal sizes that validate the condition (10) are further confirmed against all members of the *FDCAN* set, the candidate IT grades of the *IT*CAN\_s set and, as well as, the measured RE-shaft data, through the constraints, Figure 1(b),

$$\text{NS}\_{\text{CAN},\text{n}}\text{-FD}\_{\text{CAN},q} \ge \max \text{Rd}\_{\text{M}\_{\text{s}}s} \quad \forall \text{ FD}\_{\text{CAN},q} \in \text{FD}\_{\text{CAN}} \tag{11}$$

ζ=1, 2, …, j 1 ≤ j ≤ 20

$$\min \mathbf{R}d\_{\mathsf{M}\_{\mathsf{S}}} \ge \mathbf{NS}\_{\mathsf{CAN}\_{\mathsf{s}}\mathbf{n}} - \mathbf{FD}\_{\mathsf{CAN}\_{\mathsf{s}}\mathbf{q}} - \mathbf{IT}\_{\mathsf{CAN}\_{\mathsf{s}}\mathbf{s}\_{\mathsf{s}}} \forall \; \mathbf{FD}\_{\mathsf{CAN}\_{\mathsf{s}}\mathbf{q}} \in \mathbf{FD}\_{\mathsf{CAN}} \tag{12}$$

In case no member of the *FDCAN* set satisfies the condition (11) for a particular NSCAN\_n, the latter is excluded from the *BH*SG set. Moreover, in case no member of the *IT*CAN\_s set satisfies the condition (12) for a particular pair of FDCAN\_q and NSCAN\_n, validated by (11), they are both excluded from the *BH*SG set. In a similar manner, the production of the suggested Basic Shaft fits set is achieved by the following set of conditions,

$$\min \mathbf{Rd}\_{\mathsf{M}\_{\succ}s} \ge \mathbf{NS}\_{\mathsf{CAN},\mathsf{n}} \text{-- } \Pi\_{\mathsf{CAN},s,\zeta} \text{ } \forall \ \mathbf{NS}\_{\mathsf{CAN},\mathsf{n}} \in \mathbf{NS}\_{\mathsf{CAN}} \tag{13}$$

#### ζ=1, 2, …, j 1 ≤ j ≤ 20

138 Reverse Engineering – Recent Advances and Applications

Following members of the sets are then calculated by an incremental increase, *δ*, of

NSCAN\_h\_<sup>ν</sup> ≤ min*RdM\_h*  NSCAN\_s\_<sup>μ</sup> ≥ max*RdM\_s*

with the populations *ν, μ* not necessarily equal. In the relevant application example of section 6, *δ* is taken *δ=0.05mm*. Other *δ-*values can be, obviously, used depending on the case. Since both hole and shaft have a common nominal size in ISO-286 fits, the Candidate Nominal Sizes Set, *NSCAN*, is then produced by the common members of *NSCAN\_h*, *NSCAN\_s*,

 *NS*CAN = *NS*CAN\_h ∩ *NS*CAN\_s (9)

In Step (c) of the analysis, a combined qualification for the members of the *IT*CAN\_h, *IT*CAN\_s, *FDCAN* and *NS*CAN sets is performed in order to produce the two sets of *suggested* Basic Hole, *BH*SG, and Basic Shaft *BS*SG fits. The members of *IT*CAN\_h and *IT*CAN\_s sets are sorted in ascending order. For the production of the *BH*SG set, every candidate nominal size of the

NSCAN\_n+ ITCAN\_h\_<sup>κ</sup> ≥ max*RdM\_h* ∀ NSCAN\_n ∈ *NS*CAN (10)

In case no member of the *IT*CAN\_h set satisfies the condition (10) for a particular NSCAN\_n, the latter is excluded from the *BH*SG set. In order to qualify for the *BH*SG set, candidate nominal sizes that validate the condition (10) are further confirmed against all members of the *FDCAN* set, the candidate IT grades of the *IT*CAN\_s set and, as well as, the measured RE-shaft data,

NSCAN\_n– FDCAN\_q max*RdM\_s* ∀ FDCAN\_q ∈ *FD*CAN (11)

 min*RdM\_s*  NSCAN\_n – FDCAN\_q – ITCAN\_s\_<sup>ζ</sup> ∀ FDCAN\_q ∈ *FD*CAN (12) In case no member of the *FDCAN* set satisfies the condition (11) for a particular NSCAN\_n, the latter is excluded from the *BH*SG set. Moreover, in case no member of the *IT*CAN\_s set satisfies the condition (12) for a particular pair of FDCAN\_q and NSCAN\_n, validated by (11), they are both excluded from the *BH*SG set. In a similar manner, the production of the suggested Basic

min*RdM\_s* ≥ NSCAN\_n– ITCAN\_s\_<sup>ζ</sup> ∀ NSCAN\_n ∈ *NS*CAN (13)

*NS*CAN set is initially validated against all members of the *IT*CAN\_h set, Figure 1(a),

NSCAN\_s\_2 = NSCAN\_s\_1 - δ

……………………………… NSCAN\_s\_μ = NSCAN\_s\_μ-1 - δ

NSCAN\_s\_3 = NSCAN\_s\_2 – δ (7)

(8)

NSCAN\_h\_1 and NSCAN\_s\_1,

bounded by,

 NSCAN\_h\_2 = NSCAN\_h\_1 + δ NSCAN\_h\_3 = NSCAN\_h\_2 + δ ……………………………… NSCAN\_h\_ν = NSCAN\_h\_ν-1 + δ

**3.3 Sets of suggested fits** 

κ=1, 2, …, i 1≤ i ≤ 20

through the constraints, Figure 1(b),

Shaft fits set is achieved by the following set of conditions,

ζ=1, 2, …, j 1 ≤ j ≤ 20

$$\min \mathbf{R}d\_{\mathrm{M},\mathrm{f}} \ge \mathrm{NS}\_{\mathrm{CAN},\mathrm{n}} + \mathrm{FD}\_{\mathrm{CAN},\mathrm{q}} \,\forall \, \mathrm{FD}\_{\mathrm{CAN},\mathrm{q}} \in \mathrm{FD}\_{\mathrm{CAN}} \tag{14}$$

$$\text{NS}\_{\text{CAN},\text{n}} + \text{FD}\_{\text{CAN},\text{q}} + \text{IT}\_{\text{CAN},\text{h},\text{x}} \ge \max \text{Rd}\_{\text{M},\text{h}} \quad \forall \text{ FD}\_{\text{CAN},\text{q}} \in \text{FD}\_{\text{CAN}} \tag{15}$$

#### κ=1, 2, …, i 1≤ i ≤ 20

Fig. 1. Suggested Basic Hole fits qualification
