# Three-Dimensional Synthesis of Manufacturing Tolerances Based on Analysis Using the Ascending Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}, the reference surfaces, which must be in a zone of width $\Delta {\mathrm{l}}_{\mathrm{i}}^{\mathrm{j}}$, (i: the surface number, j: the machining phase number); (3) Treatment of the functional specifications by decomposing them into manufacturing specifications; (4) Formulation of inequalities from the tables already drawn; and finally (5) Solving the inequalities in order to obtain all the dimensions and manufacturing tolerances, by a calculation algorithm. The Δl method has been applied in [4,5,6,7] and the accuracy of this method has been approved. The multidirectional approach presented in Anselmetti B [8] allowed the study of a combination of defects on sloped surfaces or cones. This approach introduced the connection between surfaces machined by the same cutting tool on CNC machines (the tool offsets errors are not considered on some manufactured dimensions). The 2D statistical simulation method presented by Chep A [9] allowed the distribution of the geometric tolerances zones of square, rectangular or circular shapes, taking into account the capability of the machine. The distribution method is based on a surface decomposition of the tolerance zone of the geometrical specification taking into account the ISO dimensioning of the processing phases.

## 2. Description of the Method Used for the Synthesis and Analysis of Manufacturing Tolerances

## 3. Summary of Results Denoted from the Analysis Phase

_{i}during the phase 40.

_{i}in phase 30.

- -
- If ${\beta}_{S3/M3}^{30}+{\beta}_{H/M}^{30}$ < 0, the most unfavorable dispersion gives the following relation:$${\beta}_{S3/P3}={\beta}_{S3/M3}^{30}+{\beta}_{H/M}^{30}-{\xi}_{30}$$$$C=({c}_{4}-h)\xb7{p}_{4}$$
- -
- If ${\beta}_{S3/M3}^{30}+{\beta}_{H/M}^{30}$ > 0, the most unfavorable dispersion gives the following relation:$${\beta}_{S3/P3}={\beta}_{S3/M3}^{30}+{\beta}_{H/M}^{30}+{\xi}_{30}$$$$C={c}_{4}\xb7{p}_{4}$$

## 4. Presentation of the Synthesis Phase

_{i}of the toleranced surface S4:

- η: Influence of fixture defects,
- ξ: Influence of dispersions,
- μ: Deviation of the machined surface.

_{i}is a function of the positioning defects of the phase 40 and of the orientation defects of the phase 30. The tolerance must therefore be distributed over these two phases.

_{max}) as the other defects are estimated. Then, as before, the cumulative number of defects is compared with the imposed requirement. Conversely, the third method consists of estimating μ and ξ to allocate the largest possible tolerance ε on the machining set-up.

## 5. Analysis of Tolerances

_{1}, as a model that applies the same procedure of analysis sufficiently developed in Ayadi et al. [1].

#### 5.1. Choice of the Nominal Model of the Part

#### 5.2. Study of the Requirement E_{1}

_{1}(Figure 5). For this, it is necessary to determine the difference between the actual tolerated surface S5 and the corresponding nominal surface P5 according to the various influencing defects of the manufacturing process. This difference must be validated for all points T

_{i}of the nominal surface P5. In practice, it suffices to carry out the study at the four vertices of the rectangle, which limits the surface P5.

#### 5.2.1. Implementation of the Ascending Approach

#### 5.2.2. Study of the Phase 40

**a. Positioning of the Part**

_{1}, A

_{2}and A

_{3}) on the machined surface S1, a secondary linear support (A

_{4}and A

_{5}) on the surface S3 and a tertiary point support (A

_{6}) on the raw surface B14 (Figure 9).

**b. Influence of Machining Defects**

_{i}, the relation to be verified becomes:

- -
- The deviation of the machined surface S5 from the nominal machine surface ${M}_{5}^{40}$ (programmed surface). This deviation can be measured directly on the machine with a Renishaw probe, for example at all points Ti. This difference can be increased by a value ${\mu}_{5}^{40}$ to be fixed later.
- -
- The difference between the nominal area of part P5 and the nominal machine surface ${M}_{5}^{40}$, which must therefore be calculated to comply with the new relationship:$$\left|\overrightarrow{{T}_{i}^{N}{T}_{i}^{M}}\xb7\text{}\overrightarrow{{n}_{5}}\right|\le \text{}\frac{0.1}{2}-{\mu}_{5}^{40}$$

**Note:**there are four points T

_{i}at the four corners of the planar face. There are two sides in the tolerance zone, which actually gives eight inequalities to be respected, taking $\overrightarrow{{n}_{5}}$ and $\overrightarrow{{-n}_{5}}$.

**c. Influence of Machining Set-Up Defects**

_{j}of each support (pin) from the machine reference mark can be measured directly on the machine with a contact probe or with a comparator.

_{M40}at the point ${O}_{M}^{40}$ by the following form:

_{i}following the normal $\overrightarrow{{n}_{5}}$, caused by the machining set-up defects in phase 40, is described by the following relation:

_{j}therefore gives all the components sought.

① | $\{\begin{array}{c}{w}_{H/M}^{40}+{\alpha}_{H/M}^{40}\xb7{b}_{1}^{40}-{\beta}_{H/M}^{40}\xb7{a}_{1}^{40}=0\\ {w}_{H/M}^{40}+{\alpha}_{H/M}^{40}\xb7{b}_{2}^{40}-{\beta}_{H/M}^{40}\xb7{a}_{2}^{40}={\epsilon}_{2}^{40}\\ {w}_{H/M}^{40}+{\alpha}_{H/M}^{40}\xb7{b}_{3}^{40}-{\beta}_{H/M}^{40}\xb7{a}_{3}^{40}={\epsilon}_{3}^{40}\end{array}$ |

② | |

③ |

_{i}is written, due to the defects of the machining set-up in phase 40, as a function of the measured deviations:

**d. Influence of Part Defects**

_{1}, A

_{2}and A

_{3}. Neglecting the dispersions, the surface S1 rests on the three points of the set-up. ${H}_{1}^{40}$ and S1 are the same, so the deviation between the part nominal surface P1 and the plane ${H}_{1}^{40}$ is that of the real surface S1 relative to P1. This deviation is given by the machining of the surface S1 in phase 10.

_{i}is deducted:

**e. Influence of Dispersions**

- -
- ${\delta}_{1}^{40}$, ${\delta}_{2}^{40}$ and ${\delta}_{3}^{40}$: Deviations, respectively, on the primary supports (planar support) ${A}_{1}^{40}$,${A}_{2}^{40}$ and ${A}_{3}^{40}$.
- -
- ${\delta}_{4}^{40}$ and ${\delta}_{5}^{40}$: Deviations, respectively, on the secondary supports (linear support) ${A}_{4}^{40}$ et ${A}_{5}^{40}$.
- -
- ${\delta}_{6}^{40}$: Deviation on the tertiary support (point support) ${A}_{6}^{40}$.

_{i}of the toleranced surface.

_{i}, considering −Δ/2 ≤ δ ≤Δ/2

_{i}following the normal $\overrightarrow{{n}_{5}}$, caused by the dispersion in phase 40, is described by the following relation:

① | $\{\begin{array}{c}{w}_{D}^{40}+{\alpha}_{D}^{40}\xb7{b}_{1}^{40}-{\beta}_{D}^{40}\xb7{a}_{1}^{40}={\delta}_{1}^{40}\\ {w}_{D}^{40}+{\alpha}_{D}^{40}\xb7{b}_{2}^{40}-{\beta}_{D}^{40}\xb7{a}_{2}^{40}={\delta}_{2}^{40}\\ {w}_{D}^{40}+{\alpha}_{D}^{40}\xb7{b}_{3}^{40}-{\beta}_{D}^{40}\xb7{a}_{3}^{40}={\delta}_{3}^{40}\end{array}$ |

② | |

③ |

_{i}, according to the estimated dispersions Δ

_{i}will therefore be with the following form:

**f. Synthesis of the Phase 40**

_{i}in the direction $\pm \overrightarrow{{n}_{5}}$.

#### 5.2.3. Study of the Phase 30

_{1}, is deducted from the relation (58).

_{j}in phases 30 and 40.

_{2}to E

_{6}. The next section develops the steps of tolerances synthesis based on this analysis.

## 6. Synthesis of the Manufacturing Tolerances

#### 6.1. Qualitative Analysis of Transfers

- -
- Machined surfaces in phase n;
- -
- Surfaces that are used for positioning in phase n.

- -
- Each non-direct requirement will be decomposed into manufactured specifications that will be shown on the relevant phase drawings;
- -
- The direct requirements will be the manufactured specifications to be carried directly on the phase drawings, possibly with a reduction of the tolerance if the manufactured dimension is constrained by a transfer.

#### 6.2. Relationships Given by the Tolerance Analysis

_{1}) is applied to each functional or manufacturing requirement Ej (there are as many calculations as there are requirements). This study gives a global system of equations relating to the various geometric specifications. This system allows the synthesis of tolerances by distributing them to the different phases of the part production.

_{1}: A non-direct requirement for the groove depth between the surfaces S2 and S5, not active in the same phase. There is therefore a transfer.

_{i}of the surface S5 (calculable from the measurement of support defects ε

_{i}).

_{i}of the surface S5 (calculable from the measurement of support defects ε

_{i}).

_{i}of the surface S5 (calculable from the estimation of dispersions Δi on each mounting support).

_{i}of the surface S5 (calculable from the estimations of dispersions Δi on each mounting support).

_{2}: A non-direct requirement of position of the median plane of the groove relative to the surface S2 (primary) and S3 (secondary), which are not active in the same phase. There is therefore transfer on the three phases:

_{3}: A direct requirement of the groove width between two surfaces produced in the same phase 40 by the same tool. This dimension (which is the tool dimension) does not depend on defects of the machining set-up.

_{4}: A direct requirement for the height of the part between the surface S2 produced in phase 30 with respect to S1. There is no transfer.

_{5}: A direct requirement of the width of the part between the surface S3 produced in phase 20 and the support surface in this phase S4.

_{6}: A direct geometry requirement, which concerns only one surface, produced in phase 20. There is no influence of the machining set-up, but the flatness of the machined surface must be satisfactory.

#### 6.3. Principle of the Tolerance Synthesis

#### 6.3.1. Analysis of the Equations Term’s

_{1}and E

_{4}give the typical form of the inequalities to be respected.

_{4}):

_{1}):

_{i}of the plane face and in both directions of the normal to this face. This gives eight inequalities per requirement:

_{1}, A

_{2}and A

_{3}), a secondary linear support (A

_{4}, A

_{5}) and a tertiary point support (A

_{6}).

_{i}, y

_{i}, z

_{i}depend on the point T

_{i}where the requirement is expressed.

#### 6.3.2. Optimization of Machining Tolerance μ

_{i}at each point T

_{i}, of the deviations ${\epsilon}_{j}^{ph}$ of the machining set-up. This influence η

_{i}can be calculated at each point T

_{i}. In forecasting, it is possible to allocate a tolerance on each support ${\epsilon}_{j}^{ph}\le {\epsilon}_{maxi}$, which makes it possible to determine the maximum influence with the following relation:

_{i}, and on the geometric characteristics of the part and the part holder. If the maximum value of Δ is estimated, it is possible to calculate the deviation ${\xi}_{i}^{40}$ at each point T

_{i}in the form $\xi i=ki\xb7\Delta $.

_{i}studied.

_{i}and to the two orientations of normals n

_{1}and −n

_{1}.

_{i}.

_{i}and ξ

_{i}, which will give eight simpler inequalities.

#### 6.3.3. Optimization of the Assembly Precision ε

_{i}.

^{ph}is equal to the number of phases).

## 7. Writing of the Manufactured Dimensions

- -
- The specifications without transfer are copied directly onto the corresponding phase drawing;
- -
- For each machining phase, a reference system can be constructed on the phase positioning surfaces while respecting the order of primary, secondary and tertiary preponderance. It is preferable to use partial references or references on restricted areas to best represent the real contact areas between parts and machining fixtures;
- -
- Each transfer generated by the ascending method will be transcribed into a specification with respect to the positioning reference system in the phase: We will have a localization if the transfer relation includes a sliding term (u, v or w) or an orientation if there are only terms of rotation (α, β or γ).

_{i}of the terminal surface of the requirement, which does not belong to the machined surface. This corresponds to a 3D generalization of the concept of projected tolerance. It is therefore very difficult to relate this calculation to the classic concept of manufacturing tolerance in ISO.

## 8. Conclusions

- -
- The analysis phase is ended by writing equations that allow the analysis of the machining tolerances, and verifying the feasibility of the process according to the estimated defects;
- -
- Using the proposed synthesis methodology, it was easy to select equations that need to be respected for each requirement, write relationships defining the optimal condition expressing the deviation of the machined surface, and the optimal condition expressing the precision of the workpiece fixture;
- -
- Based on these optimizations one can write the manufacturing specifications, and prepare the drawing describing the optimal sequencing of the processing phases;
- -
- Using the phase drawing, a machinist can prepare the machine and the part fixture and by the end make the machining, which can be validated by a future experimental work.

- -
- An experimental study is needed, which aims to validate the tolerancing analysis and synthesis presented in this paper.
- -
- Analysis and synthesis of tolerance from a statistical point of view: we have developed the formulation of the problem for an analysis and synthesis of tolerance in the worst case, and a statistical approach would make it possible to resolve problems posed by very large production series more precisely.
- -
- Generalized TMT model: we presented our TMT tolerancing model by treating simple prismatic parts and using isostatic-machining fixtures with six supports. As such, a generalization study would have to be developed with more complex parts and other types of fixtures.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ayadi, B.; Ben Said, L.; Boujelbene, M.; Betrouni, S.A.
Three-Dimensional Synthesis of Manufacturing Tolerances Based on Analysis Using the Ascending Approach. *Mathematics* **2022**, *10*, 203.
https://doi.org/10.3390/math10020203

**AMA Style**

Ayadi B, Ben Said L, Boujelbene M, Betrouni SA.
Three-Dimensional Synthesis of Manufacturing Tolerances Based on Analysis Using the Ascending Approach. *Mathematics*. 2022; 10(2):203.
https://doi.org/10.3390/math10020203

**Chicago/Turabian Style**

Ayadi, Badreddine, Lotfi Ben Said, Mohamed Boujelbene, and Sid Ali Betrouni.
2022. "Three-Dimensional Synthesis of Manufacturing Tolerances Based on Analysis Using the Ascending Approach" *Mathematics* 10, no. 2: 203.
https://doi.org/10.3390/math10020203