New Approach to Integration of Form Deviations in the Analysis of Geometric Tolerances

— Tolerancing is an important step in the design process of a mechanism. There are two tolerancing approaches: tolerance analysis and tolerance synthesis. The principle of tolerance analysis is to verify the functional condition of the mechanism, for that purpose it is necessary to check the specific tolerances of each elementary part. Whereas the tolerance synthesis aim to calculate the specific tolerances of the parts forming the mechanism. The purpose of this paper is to develop a new approach of geometric tolerances analysis with integration of form deviations. For this we studied an example of assembly of two cylindrical parts (shaft and bore). The first step is to analyze the tolerances of the component parts assembly using the deviation domain method by neglecting the form deviations. The second step of this study is to analyze the tolerances with the same method but with the introduction of the form deviations, by using the discrete modal method for the modeling of surfaces with form deviations and applying a new algorithm developed for the simulation of our approach. Finally, we compared the two studies with and without form deviations through the calculation of a non-compliance rate (TNC)


IV. DISCRETE MODAL DECOMPOSITION METHOD (DMD)
The DMD is based on the vibration theory of discretized mechanical structures. Each natural mode of vibration defines a particular geometry. These modes are used as parameters defining the form deviations of a surface [7]. These modes of vibration are calculated from equation 2:

M.
K. u 0 Where K and M are matrices of rigidity and mass, and u is the displacement vector. The solution of equation (2) provides a linear system whose solutions are the natural mode of vibration Qi corresponding to the pulsation i.
Each natural vector is normalized according to the infinite norm so that ‖Q ‖ 1.
To generate surfaces with form deviations, it is necessary to discretize the surface in finite elements and to compute in a first time the natural modes of this surface, then generate the modal coefficients. Each surface is calculated with the following equation: S Q. m 4 Where Q is the matrix of the natural modes of the surface and m is the modal coefficients vector.

V. TOLERANCING OF AN ASSEMBLY (ASSEMBLY OF A SHAFT WITH ITS BORE)
The example studied is an assembly of two cylindrical parts. The functional condition to verify is the assemblability of the shaft with the bore, which is represented by a geometric tolerance of coaxiality. The assembly and definitions drawing of parts are illustrated in figure 2.
Hypotheses • The parts are rigid.
• The study of parts concerns only the functional surfaces The part (1) is considered as a reference part for the assembly, so the deviation domain of the cylindrical surface (A1) is zero: E1A = 0.
 Study of the part (2): Shaft The surface (B2) of the part (2) is a reference surface of coaxiality tolerance of the surface (A2), so the deviation domain of this surface is zero: E2B = 0 The surface (A2) is a cylindrical surface toleranced in coaxially with related to the surface (B2), so the deviation torsor of this surface is of the following form: To respect the coaxial tolerance assigned to the cylindrical surface (A2), the displacement of the points O1 and O2 along the two axes X and Y must not exceed the tolerance interval divided by 2 (to / 2).
The deviation domain of this surface in the center of the cylindrical link is of the following form: The graphical representation is of the following form:  Study of the functional condition of the assembly It is considered that the tolerance of coaxiality represents the functional condition of the assembly, so for the assembly to be functional it suffices to respect this tolerance during the manufacture of the parts.
Calculation of natural modes of the studied surface The two surfaces in contact in our assembly are cylindrical surfaces. The natural modes of this surface are calculated. To simplify the study, we consider that the cylindrical surface is represented with its axis with the deformations caused by vibrations. The following figure shows some natural modes of a cylindrical surface. Generations of surfaces with form deviations Each surface is modeled with a set of nodes that undergo displacements along the y axis, the displacements of all the nodes of all the natural modes are normalized so that the displacements are between "-1" and "1".Then we generated for each natural mode 20 random values according to the normal law between 0 and 1. In our case each surface is composed of 5 nodes. The calculation is done using the following equation using "Excel ": S1.1 S1.2 ⋮ S1.5 -S1.1 : Displacement of node 1 of surface 1 -Q1.1 : Value of mode 1 for node 1 of surface i -m1.1 : Random coefficient for mode 1 of surface 1 We generated 20 random surfaces form deviations based on the natural modes of vibration of the cylindrical surface. The following figure shows the calculated surfaces with form deviations. -Verification of the conformity of the part taking into account the form deviations: for the part to be accepted, the control parameters must stay within the tolerance interval. -Representation of the conforming parts with the consideration of form deviations: each piece i conforms is represented with a blue point of coordinates (txi, ryi). -Verification of the conformity of the parts without taking into account the form deviations and which are rejected with the taking into account form deviations: in order for the part to be conforming, it is necessary that the parameters of control stay within the tolerance interval without form deviations. The parts represented with blue points are accepted with the taking into account of the two types of deviations: the coaxiality and the form deviations .The parts represented with red stars are defective parts if we take into account the form deviations, whereas they are accepted if we neglect the form deviations. To illustrate the influence of the introduction of the form deviations on the tolerance analysis with the domain method, we calculated the rate of non-conformity which is of the order of 5%.
VI. CONCLUSION In this paper we studied through an example of assembly of two cylindrical parts, the geometric tolerances analysis with and without taking into account the form deviations. The first part presents the tolerancing without taking into account form deviations, by using the deviations domains method with the worst-case approach. The second part presents tolerancing with consideration of form deviations. In order to model these form deviations we used the discrete modal decomposition method. Then we developed a Monte-Carlo statistical simulation for the tolerance analysis based on the new developed algorithm. Finally, the result is represented with a graph to facilitate interpretation, and then we calculated a non-conformity rate which illustrates the difference between tolerancing with and without taking into account form deviations.