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Procedia CIRP 10 (2013) 178 - 185

12th CIRP Conference on Computer Aided Tolerancing

Form errors impact in a rotating plane surface assembly

J. Grandjeana*, Y. Ledouxb, S. Sampera, H. Favrelièrea

a Université de Savoie, Laboratoire SYMME ,BP 80439, 74944 Annecy le Vieux Cedex, France bUniversité de Bordeaux, I2M UMR 5295, Esplanade des Arts et Métier, 33400 TALENCE, FRANCE

Abstract

This paper focuses on surface defects of mechanical joint composed of two plane surfaces of two carters assembly. The originality of the proposed study is to manage both form errors and sliding surfaces complexities in a Computer Aided Tolerancing process. The assembly procedure is introduced in a tolerancing analysis process in order to assess designer's tolerance choices. It's also considered the angular orientation of the carter's plane surface, corresponding to the mobility of the assembly. The domain of relative positions according to this positioning parameter is computed. This one, called "mobility precision domain", is defined for a particular sliding assembly.

© 2013 The Authors.PublishedbyElsevierB.V.

Selection and peer-review under responsibility of Professor Xiangqian (Jane) Jiang Keywords: form errors; mobility; precision; 3D assemblies

1. Introduction

The continuously research of product better performing, more energy-efficient, and more cost-effective, added to the continuous search for innovation have given rise to an increasing need for understanding the influence of geometrical defects of parts and develop means to measure these geometrical defects and to define a particular language to specify them.

Face to these challenges, geometric tolerances have been gradually introduced with the aim at providing a more comprehensive way for defining allowable geometrical variation of product subject to functional and technical requirements. Based on geometric tolerances, it is possible to describe different type of variations related to shape, position and orientation of geometrical features. Even if geometrical specifications are widely recognized as a key element to ensuring a suitable level of quality for features, products or assemblies, it remains some restrictions based on

* Corresponding author. Tel.: +33 (0) 4 50 09 65 97; fax: +33 (0) 4 50 09 65 43 . E-mail address: julien.grandjean@univ-savoie.fr.

intrinsic assumptions as no consideration of form errors of surfaces and stiffness of parts.

In order to give an answer to these fundamental restrictions, some authors proposed to introduce local deformations into the geometrical model of parts when mechanisms are subject to thermal loads [1, 2]. Others study aims at considering the displacements of joints when the mechanism is subjected to important mechanical loads [3-5]. In order to take into account waviness and roughness, [6] gives multi-scale solutions in FEA solvers limited to local analyses. Concerning the integration of surface defects of mechanical joints, it could be cited the recent works [3, 7-9,]. In [10] the authors show how they study a sliding assembly having form errors by computing contact configurations of NURBS surfaces. Our method seems to be more flexible. In [11], the modal parameter is presented as a new method allowing to define form errors of any surface.

The influence of surface defects could have a significant influence in many cases [12] even if the different geometrical specifications of shape and position are well defined. In a general case, even if the position tolerance is bigger than two times the form error one, we could obtain some no conform assembly

2212-8271 © 2013 The Authors. Published by Elsevier B.V.

Selection and peer-review under responsibility of Professor Xiangqian (Jane) Jiang

doi:10.1016/j.procir.2013.08.029

although all functional specifications have been respected.

This last point corresponds to one of the main focuses of this paper. To do so, we developed a framework to integrate such form defects into the geometrical specification of product. To illustrate the approach and the associated tools, we use a classical assembly composed of two housings linked through a ball and cylinder joints added to a planar joint. Such an assembly is classically used for pump bodies or turboshaft engine crankcases. The main functional requirement is to precisely position part 2 compared to part 3. Both parts position respectively the point A and E ensuring the localization of the rotor into the crankcases (see Fig. 1 and 2).

In this study, it is shown on one hand how to consider form defects into the tolerance analysis process and one the other hand how to control their impact on the parts positioning accuracy. Various applications of this approach can be considered as:

• During the design of the mechanism, the quantification of the impact of types of defects on the modification of the positioning of the functional surfaces. It may be possible then to specify the amplitude of form defect.

• During the industrialization phase as a tool to select the most suitable means of production according to their form defects types.

• During the assembly phase, according to the form defect type of parts to be assembled, it could be possible to define a pairing strategy.

Fig. 1. Simplified representation of the mechanism.

( 1 ti A B

EJ t2

plane C2

Fig. 3. Simplified representation of the specification of the C1 annular flat surface

Fig. 2. Detail of the geometrical specifications of part 2.

-005'""1 (10-5 d)

Fig. 4. Representation of all possible displacements of the surface C2 according to the geometrical specification.

In this paper, the authors focus this approach in the design step. The goal is then to determine the maximum possible range of variations depending on the types of form defects. The main purpose is to define all the extreme position of the crankcase 2 compare to the crankcase 3 according to the set of angular positions corresponding to a set of discrete positions defined by the number of screws used for holding in position.

To do so, different kind of form, position and orientation defects are studied. The base of defect types is driven by the "modal composition" [11] method from natural eigenmodes of initial surface. The different examples allow quantifying the influence of the form defects of the functional surfaces composing the mechanical joints.

2. Functional requirement

According to the geometrical specification of location imposed by the designer, it is possible to determine all positions that plane C2 could have. If it is only considered the extreme position, it is possible to deduce the deviation domain. This representation and the associated computation are derived from previous works (for more details, see [15, 9, 14, 13].

Let consider a point Pi belonging to the planar surface C2. In order to respect the specification of location, all Pi points must remain within two parallel planes, see Fig. 3. The distance between these two planes corresponds to the tolerance value t. They are on both sides of the nominal plane. In this study the value of t corresponds to 80|im and the inner and outer radii of the crankcase are 90mm and 120mm.

It is possible to determine all allowed displacements of Pi through the relation (1) where 5pi corresponds to the translation vector of pi.

p. z ^ t (1)

The pi coordinate could be defined by the relation (2) as a function of the external radius r of the part 2.

Pi = (cos(0) x r, sin(0)x r, A) (2)

Thus, all the allowed displacements of pi that are expressed in the frame (O, x, y, z) are determined according to eq. 3. With p corresponding to rotations of C1 planar surface along x and y axes; So is the translation component (noted Tx, Ty, Tz) of the small displacement of C1.

dpi = So + opi x p (3)

If it is only considerate the displacement along z (corresponding to the projection), it could be deduce the relation 4.

Spi.z =Tz + cos(0)xrxRy - sin(9)xrxRx (4)

And according to eq. 1, the geometrical specification could be expressed by the relation (5).

Tz + cos(Q) x r x Ry - sin(6)x r x Rx s^t1/2

Such as 9 e [0,2k] (5)

This inequality determines the set of possible value of the geometrical parameters Rx, Ry and Tz corresponding to the three measurable small displacements (i.e. translations and rotations) of the surface C2. The geometrical representation of this domain is displayed in Fig. 4.

3. Form defects generation

One of the most common defects that could be occurred in a lathe operation corresponds to the local deviations of the plane surface of the crankcase due to the gripping force of the chuck composed with 3 concentric jaws. As it can be seen in the Fig. 5, the

gripping loads are responsible of the apparition of tri-lobed defects (composed of three lobes onto the front face of the part). In this example, the waviness range is about 0.014mm.

8 6 4 2

Range (Mm) 0 -2

Fig. 5: Measurements of local deviations associated with classical lathe gripping means.

Such periodic defects, according to their angular orientation could have a positive or negative effect depending on the orientation of assembled parts or the rigidity of the final assembly.

Looking at the measured surface on Fig. 5, it can be observed that the form defect does not only correspond to a periodic waviness defect (one lobe every 120°), but contains other combined defects. Within this context, the objective of the present work is to validate the process leading to estimate the influence of plane surface form defects on the accuracy of the final assembly. This would avoid using real surfaces that require machining and detailed surface measurement. In fact, the use of synthetic surfaces represents a major advantage if the aim is to perform a parametric analysis of the mechanical joint behaviour in order to aid to their design to define and quantify the geometric tolerances. In this work, the synthetic surface is built by a composition of natural eigenmodes of the initial surface computed by a Modal Discrete Decomposition (MDD).

3.1. Definition of the modal discrete decomposition

The MDD is based on the vibration theory of discrete mechanical structures (FEA). Every eigen vibration modes defines a particular geometry and these modes are combined to parameter synthetic surface [11-12]. The eigenmodes of the annular flat surface are obtained by the resolution of the dynamic conservative equilibrium given by the equation 6. This equation is a function of the mass (M) and the stiffness (K) matrices and the nodal displacement vector u.

+ K-u = 0 (6)

Angular value (°)

The equation 6 provides a linear system where the solution is the eigenmodes Qt corresponding to the pulsation (Oj.

where I is the identity matrix.

In the case of free boundary conditions of the annular flat surface, the resolution of the equation 7 leads to find for the three first (we have removed the three translation modes because they are not useful here) modes Qj to Qj, the rigid body modes (rigid displacement of the surface), and for the others, Qt (i= 4, n), the vibration modes of the surfaces. In the proposed procedure, every eigenvector is normalized according to the infinity norm

so that llfil = 1

Fig. 6. The seven first natural modal shapes of the surface

Fig. 6 illustrates the seven first modes obtained with this approach. The seventh modal shape is characteristic of the periodic waviness defect seen in section 4.2.

3.2. Generation of synthetic surface

On the studied application, the annular surface is generated with a radius r ranging from 90 to 120mm. The discrete surface is built with 9800 shell elements composed of four nodes (with N=NrxN0, Nr=71, N=140). All elements have three degrees of freedom (two rotations and one translation). From this discrete surface, all synthetic surfaces (represented by the nodal displacement vector V) are computed (eq. 8) through the product of the matrix (Q) of the set of modes Qi(i=1,n) and the modal coordinates (m).

V = Q • m

Such as Q is a N x n matrix of eigenmodes Qi and m is the vector of modal coefficients mi (i=1,n).

Since Qi are unit vectors, a metrics can be attributed to mi coefficients. Fig. 7 illustrates the surface defect generation.

m,.Q, m,.Q, v

Fig. 7. Illustration of the surface defect generation

mi (mm)

Fig. 8. Modal coordinates of a synthetized of a surface

When we measure surfaces and we calculate their modal decomposition, we always get a natural decrease of the modal coordinates. This behavior is well known in signal processing and can be used to simplify the measured signal. We use a similar method in space dimensions. We have built a form error synthesizer having this property. In this one, we compute a random shape of a surface according to the decreasing law. In Fig. 8, we can observe the modal coordinates of a synthetized surface.

3.3. Assembly analysis

The aim of this analysis is to define the position of a part compare to the other by knowing the form errors of the surfaces being in contact. This method has been published in [3] in details in static case of planar joint. In this paper, the approach is applied on an industrial case considering the angular mobility of the parts composing the assembly.

We make the assumption that the assembly is rigid. Thus there is only three points in contact between the two surfaces. They define the relative position frame of the assembly. The three other mobilities are given by the centering of the surfaces and the relative rotation angle. This one can change and we compute each assembly as if the assembly process would have a uniform law for this parameter.

4. Results

Several pairs of surfaces with defects are generated. For each of the two surfaces, we calculate the Small Displacement Torsor (SDT) by considering a rigid positioning on the plane z = 0. Next, we consider the assembly of the two surfaces using the sum-surface concept. For that the surface 1is considered as a mobile surface and the surface 2 as fixed to calculate the SDT of the assembly for several angular positions of the surface 1.

4.1. Examplel: Surfaces with no form error

In this first example, we consider that t1=0,05mm and t2=0. The surfaces thus generated have a position error (translation and orientation) but do not have form error (they are flat). Fig. 9 (a) shows the surface 1, Fig. 9 (b) shows the surface 2, and Fig. 9 (c) represents the sum-surface of surfaces 1 and 2 for this angular configuration of assembly.

-0.05 0 0.05

Tz (mm)

-4 -2 0 2 4

Ry (rad) x 10-4

SDT of the Surface 1

Rx = 4.99E-03 rad Ry = -1.25E-04 rad Tz = 4.99E-03 mm

SDT of the Surface 2 SDT of the Sum-surface

Rx = -4.26E-05 rad Rx = -1.37E-04 rad

Ry = 1.30E-04 rad Ry = 5.02E-06 rad

Tz = 1.83E-03 mm Tz = 6.83E-03 mm

Fig. 10. SDT of an assembly (a) in (Rx,Tz) plane; (b) in (Rx,Ry) plane.

If we compare the SDT of the two surfaces 1 and 2 compared to the sum-surface of the surfaces (Fig. 10), we conclude that the positioning of the assembly is given by the SDT named SDTsum-su,face defined below:

SDT = SDT + SDT

sum-surface ^JIS1 surfl -t surf2

Fig. 9. (a) Upper surface 1, (b) Lower surface 2, (c) Sum surface

We then calculate the Small Displacement Torsor (SDT) [14, 15] that represents the three translations and the three rotations set of displacement [3] for each surface 1 and 2 and the sum-surface. Here we will not use three on the six components of each SDT because we study a planar joint. Thus each SDT is represented by a dot in a 3D domain (Tz, Rx, Ry) where we represent also the domain of all possible displacements (deviation domain). This one is closed by a surface that is the limit values of the allowable SDT.

This eq. 9 can be used for assemblies where form errors are less important than orientation errors, to predict for example the total error of the final assembly.

-0.05 0

Tz (mm)

-4 -0.05

2\* 0 -2 -4

-2 0 2 4

Ry (rad) x 10-4

Tz (mm)

2 x 10

Ry (rad)

Fig. 11. Set of possible SDT of assemblies by rotations (a) in (Rx,Tz) plane; (b) in (Rx,Ry) plane; (c) in (Rx,Ry,Tz) plane.

To consider other angular configurations of the assembly, it now performs a rotation of the surface 1 around the axis z by considering the surface 2 fixed. For each rotation of the surface 1 of an angular step of 2jc/30 (for a total of thirty different positions), we calculate the sum-surface of the two surfaces. This results in thirty sum-surfaces representing the thirty angular positions of the surface 1 on a surface 2. The calculate SDT for each configuration are shown in Fig. 11. We selected thirty different positions to see the area of displacements, in a real case, the possible angular positions are determined by the number of screws and nuts used to assembly the crankcase.

In the (Rx, Ry) plane, Fig. 11 (b) we show that the SDT assembly is a circle whose center is the center of the SDT of the surface 2. The radius of the circle in this plane is equal to the distance [origin, SDT (surface 1)]. The Tz translation of all SDT is constant regardless of the angular position of the surface 1 (Fig. 11 (a)). As

shown in the equation 9,

Tzsmfi+Tzsurf2.

Tzsum-surface is equal to

4.2. Exemple2 : Surfaces with orientation andform errors

x 10 42-

-0.05 0

Tz (mm)

-2 0 2 4 Ry (rad) x 10-4

-44 -0.05

0.05 4

Tz (mm)

Fig. 13. (a) SDT(Rx,Tz); (b) SDT(Rx,Ry); (c) SDT(Rx,Ry,Tz)

In this example, we set t1=0,05mm and t2=0,025mm for the generation of surface defects. The pairs of surfaces thus generated have position and form errors. To simulate the three lobes onto the front face of the part caused by the gripping loads, the seventh modal shape has been magnified by a factor of 5 during the generation of defects. The surfaces thus generated are visible in the Fig. 12 below. In Fig. 13 are shown the SDT of the assembly for the different angular positions of the surface 1.

The addition of form errors has distorted the circle of SDT in the plane (Rx, Ry). Similarly we see that the translation Tz assembly and this is no longer constant and have fluctuations of the order of 0.006 mm.

The shape described by the SDTs of all assembly configurations is also composed by three lobes. That can be explained by the same major defect caused by the gripping loads on both surfaces, which allow the fitting of the two surfaces in some surface orientations.

4.3. Exemple3 :Surfaces with orientation andform errors

In this example, we keep the set t1=0,05mm and t2=0,025mm for the generation of surface defects. But only the second surface has the three lobes defects caused by the gripping loads. The two surfaces are shown in Fig. 14.

Fig. 12. (a)Upper surface 1, (b) Lower surface 2

Fig. 14. (a) Upper surface 1, (b) Lower surface 2

The Fig. 15 below shows the SDT of the assembly for these two surfaces. Three calculated STD torsor are outside of the domain of all possible displacements. This possibility has been shown in a previous work [12] by the study of the influence of form errors. Although both surfaces are conform in a metrological point of view (i.e.

respecting the geometrical specifications), the SDT torsor of the assembly can be outside the deviation domain making it no conform.

-0.05 0

Tz (mm)

-2 0 2 4 Ry (rad) x 10-4

-4 -0.05

0.05 4

Tz (mm)

2 x 10

Ry (rad)

Fig. 15. (a) SDT(Rx,Tz); (b) SDT(RxRy); (c) SDT(Rx,Ry,Tz)

As the defects are not the same type, the two parts cannot fit together as easily than in the previous example. That's why the fluctuations of translation Tz are less important, around 0.003 mm.

This approach allows calculating the optimal angular position of the surface 1 that will minimize, depending on the functional need, the error of orientation of the assembly or the error of translation. More importantly, we can see with this example that it ensures conformity of an assembly by providing the best positioning between the two surfaces.

Thus, some costly components which would require pair parts could be replaced by an optimized assembly process by including an adapted measurement step.

5. Conclusion

We proposed in this paper a method to analyse the influence of form errors on a rotating plane assembly. This analysis has been made with synthetized surface having different value and types of form errors and position defects. To generate these synthetized surfaces, the modal discrete decomposition is used. Based on these surfaces, a procedure has been developed to perform the assembly of such a surface by considering several angular position of surface along the normal axis of the planar joint. The analysis of the conformity is

made through the specification domain built in a 3D reduced SDT space. One of the main results of this work concerns the ability to quantify the evolution of the surface in function of the angular position and then identify the optimal position to respect the functional specification. The procedure corresponds to a useful tool in case of pairing assembly process where the selection of the best combination of parts can be determined automatically in function of the measured defects on parts.

It is also possible to draw on this approach to select the optimal process to use. Since all manufacturing process generate particular type of defects, it is possible to identify the main spectral decomposition of defects into the modal basis. By comparison between these different spectra, it is possible to select the most appropriate process. This quantification could be based on stochastic analysis for instance.

This initial procedure could be completed by different points. First, the integration of deformations could be introduced. The local deformations of contact surfaces could be addressed for instance through Hertzian contact model or mechanical behavior of surfaces (elasto-plastic models). Then, the deformation of the structure subject to external loads could be introduced for example by an estimation of the stiffness matrix of the parts.

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