Evaluation of Manufacturing Process based on the Geometric Variation Model in Multi-station Machining Processes

. The objective of this paper is to explore the evaluation method of manufacturing process to verify its effectiveness based on the limitation of the variations which occur in multi-station machining processes. Firstly, the manufacturing process of a mechanical part is considered as a mechanism mainly consisted of machine-tool, part-holders, machined part, and cutting tools; And small displacement torsors (SDTs) are applied to describe all deviations in the manufacturing process, including the variation deviations of the machined surfaces of a part with regards to their nominal positions, the gap deviations associated to each joint between two contact surfaces, etc; Then, the 3D manufacturing variation model is established based on the relations between the machining feature variations and the functional tolerance requirements to realize the evaluation of manufacturing process. Finally, an application example is given to illustrate the proposed method.


Introduction
Multi-station machining process is the most common, most important and most difficult to control in the manufacturing industry. The research on evaluation method of manufacturing process based on 3D manufacturing variation model for multi-station machining processes plays an important role in estimating the geometrical and dimensional quality of manufactured parts, optimizing the process route of products, generating robust process plans, and eliminating downstream manufacturing problems. In order to consider the influence of geometric process variations, Bourdet and Ballot [1] proposed a three-dimensional variations model by using the small displacement torsor (SDT) to model geometric deviations in manufacturing process. Based on the concept of SDT, Legoff et al. [2] put forward a method for performing tridimensional analysis and synthesis of machining tolerances. Villeneuve et al. [3] have proposed a three-dimensional model on manufacturing tolerancing for mechanical parts in which the SDT concept is used to model the machined parts, part-holders, and machining operations. Louati et al. [4] proposed a machining tolerancing method using SDT theory to optimize a manufactured part setting. Abellá n-Nebot et al. [5] analyzed two 3D manufacturing variation models, the stream of variation model (SoV) and model of the manufactured part (MMP), in multi-station machining systems and compared their main characteristics and applications. Furthermore, Laifa et al. [6] presented a 3D formalization of manufacturing tolerancing which associates the concept of SDTs, the functional constraints, and manufacturing process capability. This approach enables the evaluation of manufacturing process by limiting the variations which occur at the various production setups. Despite some research achievements have been made in this related field, there are still many open issues to be explored. This paper focuses on 3D modeling of the geometrical variations in the processes of part machining and exploring the evaluation of manufacturing process by using the SDT and its transfer formula.

Small displacement torsor and its transfer formula
It is generally known that any variation of a geometrical feature from its nominal position can be characterized by a SDT with three rotation components (α, β, γ) around x, y, z axes and three translation components (u, v, w) along x, y, z axes, respectively [7]. The SDT , which synthesizes the position and orientation of an associated feature Pi relative to its nominal feature at a given point O i in a local reference frame R i , can be expressed as: For a special case, the variation torsor of a plane feature Pi with regard to its nominal plane can be expressed as: where point O i belongs to the plane and R i is a local reference frame whose axis z i is the normal of the plane, capital U represents the undetermined component in the expression of a torsor.
To calculate operations on these torsors, the following two properties are defined [3]: point O i in a local reference frame R i , this SDT at a given point O expressed in the global reference frame R 0 , will become: If there is only translation transformation between R 0 and R i , the above transfer formula will be simplified as:

Modelling of manufacturing variations
The manufacturing process of a part is generally composed of different set-ups, each set-up is regarded as a mechanism mainly consisted of machine-tool, part-holders, machined part, and cutting tools. We suppose that the geometrical variations in the machining process are small enough to be modeled with SDT.

The SDT chain in the machining process
In order to model the whole manufacturing process, three types of torsors need to be defined: the global variation SDT; the deviation SDT; the gap SDT.

Modelling of the geometrical variations
The manufacturing process of a part generally consists of several different set-ups, and we will discuss how these geometrical variations are transferred between set-ups. It is assumed that torsor a b /P P τ can represent the functional tolerance between two machined surfaces P a and P b of part P, and its expression is as follows: Thus, the relationship between functional tolerances and various variations in each set-up can be established to reveal how geometric variations are transferred between set-ups. This relationship can further guide the process engineers to evaluate the manufacturing process.

Evaluation of the manufacturing processes
In order to verify the effectiveness of the manufacturing process, we will discuss the manufacturing process evaluation method that takes into account the limitations on the manufacturing variations by the tolerances related to the functional requirements.

Parallelism tolerance requirement
As shown in Fig. 1, the parallelism tolerance of surface P 7 related to datum A (P 1 ) indicates that the actual plane P 7 shall be contained between two parallel planes Tpa apart which are parallel to datum plane A. One uses the torsors to model the variations of machined plane P 7 and datum surface A (P 1 ) with regard to their nominal surface in the global reference frame R 0 . And according to Eq. (8), one gets: As shown in Fig.2, the variations of toleranced surface P 7 relative to datum surface P 1 is defined by the displacement of any point of the tolerance surface M P7 compared to the point corresponding M P1 . Since the displacement depends only on the rotation variations, it can be calculated as:

Fig. 2. Variations between toleranced surface and datum surface
To satisfy the parallelism tolerance requirement, one gets: is the normal vector to the datum plane A. By neglecting the higher-order terms beyond the first order, inequality (12) becomes:

Perpendicularity tolerance requirement
As shown in Fig.1, the perpendicularity tolerance of surface P 8 related to datum surface B (P 7 ) indicates that the actual surface P 8 shall be contained between two parallel planes Tpe apart that are perpendicular to datum plane B.
One uses the torsors to model the variations of machined surface P 8 related to datum surface B (P 7 ) in the global reference frame R 0 : As shown in Fig.3, the variations of toleranced surface P 8 relative to datum surface P 7 is defined by the displacement of any point of the tolerance surface M P8 compared to the corresponding point M P7 of the situation surface perpendicular to the datum surface P 7 . Since the displacement also depends only on the rotation variations, it can be calculated as: /P P /P P P -β α n is the normal vector to the situation plane perpendicular to datum surface P 7 . By neglecting the higher-order terms beyond the first order, inequality (16) becomes:

Application Example
This section presents a mechanical part (see Fig. 4) to demonstrate how the proposed method can be used for evaluating the manufacturing process. The functional requirements which are transferred are the parallelism of plane P 7 with respect to datum A on plane P 1 , the perpendicularity of plane P 8 with respect to datum B on plane P 7 as described in Fig. 4. And Figs. 5-7 show the machining process of this part, which consists of three set-ups performed on a numerical control (NC) machine-tool.

Set-up 10
In set-up 10, there are no positioning surfaces. As shown in Fig. 5, the machined surfaces are marked as P i , the local reference frame R i (O i , x i , y i , z i ) for each machined surface is defined as: Axis z i is normal to surface P i pointing towards the outside of the entity; Origin O i , axes x i and y i of the reference frame belong to surface P i . In the global reference frame R 0 (O, x 0 , y 0 , z 0 ) of Fig. 5, we have:

Set-up 20
In this set-up, surface 1 of the part is machined. As shown in Fig. 6 (a), the machined part is positioned on a plane (main positioning surface H 4 /P 4 ), a cylindrical surface with radius r c (2 nd positioning surface H 2 /P 2 ) and a spherical surface whose radius is r s (3 rd positioning surface H 6 /P 6 ) in an isostatic set-up. The global SDT 20 P/R τ of the machined part can be obtained by combining the torsors associated with joints between the part and the part-holder. Here, the part-holder support points are also marked as O i in a local frame R i (O i , x i , y i , z i ), i∈{2, 4, 6}.   Fig.5), the rotation matrices i , 0 R from R 0 to R i respectively are: And considering the properties (3) and (4) Considering the hierarchy of the part/part-holder positioning in an isostatic set-up, the components of

Set-up 30
In set-up 30, surfaces 7 and 8 are machined. As shown in Fig.7 (a), the part is positioned on a plane (main positioning surface H 4 /P 4 ), a cylindrical surface with radius r c (2 nd positioning surface H 1 /P 1 ) and a spherical surface with radius r s (3 rd positioning surface H 6 /P 6 ) in an isostatic set-up.

Respect of the functional tolerances
According to Section 3, we establish the relationship between the functional tolerances and the geometrical variations in the manufacturing process to evaluate the validity of the manufacturing process.
 Consider the parallelism tolerance requirement between surface P 1 and P 7 . The variation torsor of surface P 7 relative to surface P 1 (datum surface) is:

} {τ
The geometrical variation between tolerance surface P 7 and its nominal position related to the datum plane P 1 is defined by the displacement of any point of P 7 compared to the corresponding point of P 1   Consider the perpendicularity tolerance requirement between surface P 7 and P 8 . And this variation also depends only on rotation variations. It needs to check all the points of the toleranced surface are located in the spatial region between two parallel planes spaced with a distance of the perpendicularity tolerance (theoretically perpendicular to the specified datum plane). The variation of toleranced surface P 8 relative to datum surface P 7 can be written as:

Conclusions
Our research focuses on the development of 3D manufacturing variation model and verifying the validity of manufacturing process in multi-station machining processes. In multi-station machining process, each machining set-up is regarded as a single mechanism, and SDTs are employed to express the geometrical variations of parts caused by the machining operations and positioning dispersions during the successive machining set-ups. The SDT chain can be established based on the link between the process planning and functional tolerances, which makes it possible to evaluate the manufacturing process. The proposed method is illustrated by an example.