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Procedia CIRP 43 (2016) 58 - 63

www.elsevier.corr/locate/procedia

14th CIRP Conference on Computer Aided Tolerancing (CAT)

Assembly Error Calculation with Consideration of Part Deformation

Liu Tingb,Cao Yanlonga,b*,Wang Jingb ,Yang Jiangxinb

a. State Key Laboratory of Fluid Power and Mechatronic Systems, college of Mechanical Engineering, Zhejiang University, Hangzhou, 310027, China b. Key Laboratory of Advanced Manufacturing Technology of Zhejiang Province, college of Mechanical Engineering, Zhejiang University, Hangzhou , 310027,

China;

* Corresponding author. Tel.:+86-571-87953198; fax: +86-571-87951145. E-mail address: sdcaoyl@zju.edu.cn.

Abstract

Traditional assembly do not consider mechanical deformation based on the rigid body hypothesis, while assembly precision has close relationship with part' deformation in practice. In this paper, temperature, gravity and working load are taken into account to calculate the deflection occurred in assemblies. Both manufacturing error and deformation error are taken into consideration to establish deviation calculation model based on Jacobian-torsor model. A tailstock is taken as an example to verify the feasibility of the proposed method. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of the 14th CIRP Conference on Computer Aided Tolerancing

Keywords: deformation; temperature; gravity; working load; assembly error

1. Introduction

A mechanical assembly consists of two or more components or subassemblies. Owing to variation in manufacturing, it is impossible to completely avoid variations in component dimensional and geometrical characteristics. Also mechanical products are easily affected by the working environment in the assembly process. Load, temperature and gravity will bring about part' deformation, increasing the product assembly errors and affecting the assembly quality. An effective and rapid assembly error calculation method taking both part machining errors and deformation errors into consideration will provide a significant guide for reasonable and economical accuracy in design.

Many different approaches for calculating assembly errors have been developed over the years. Bensheng Xu et al. [1] choose to use the relationship between mechanical finger's differential motion and joint's small displacements to establish the model of tolerance accumulation; Tang and Davies [2] present a complete matrix tree chain method for determining the accumulated tolerances and the working dimensions. Paul et al. [3] estimate the tolerance by the calculation of the tolerance sensitivity of critical assembly features with respect to each source of dimensional variation in the assembly. C. LU et al. [4] propose a concept called sensitive tolerance to evaluate the assemblability according

to its assembly sequence. Besides, the Jacobian [5], the T-Map model and the deviation domain model [6-7], the direct linearization method (DLM) [8], the torsor model [9], the matrix model [10], the vector loop model [11] and the unified Jacobian-torsor model [12] have been presented successively. These mentioned tolerance calculation processes assume that the manufacture parts are rigid without consideration of deformation in assembly.

In actual working condition, temperature, load and part gravity may result in changes in dimension, angle and shape, then influencing the matched space position and final system reliability. So there is a great need to consider the deformation in design stage.

So far, several researchers have discussed the tolerance analysis integrated the part elastic deformation and displacement. S. Samper et al. [13] present four models in that allow elastic deformations of mechanisms in tolerancing. Pierre et al. [14] have integrated thermos-mechanical strains into tolerance analysis, where they have used finite element method to determine the strains. G. Jayaprakash [15] proposes an optimal tolerance design method for mechanical assembly considering thermal impact, and further takes both thermal and inertia impact into account. NSGA II and finite element are used to obtain proper component tolerance values [16]. Benichou et al consider thermal expansion of parts integrated within functional tolerancing [17]. As demonstrated in [18-19] a wider set of operating factors can

2212-8271 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of the 14th CIRP Conference on Computer Aided Tolerancing doi: 10.1016/j.procir.2016.02.007

be included in computer-aided tolerance analysis such as thermal, wear and pressure. Rotating machinery in boring processing, the static deformation of the spindle shaft is calculated by finite element method [20]. Tolerance allocation in assembly with time-variant deviation induced by mobility and wear is performed to optimize the tolerance specification [21]. FE simulation as a virtual tool is used to calculate actual deformations in a designed mechanical part due to all of its service loads [22]. Weiming Zhang et al extends and modifies the Jacobian-torsor tolerance model in actual working condition, quantitatively expresses the impact of actual working condition on working performance and tolerance design [23]. The methods proposed above either only suitable for deformation in special conditions without commonality or concerns no relation to accumulated assembly error, Jianyong Liu [24] puts forward a computational method for assembly error with consideration of parts deformation. Nevertheless, analysing dimensional accuracy in deformation situation only, invalid for the relevant location and form error.

This paper presents a new method to establish deviation propagation and calculation model considering manufacturing error and deformation error simultaneously. Deformation error results from the effects of temperature, gravity and working load on assembly part. Jacobian-torsor model is adopted to obtain comprehensive error.

The organization of the article is as follows. The factors that have an impact on part deformation are discussed in Section 2. Section 3 proposes the theory and procedures for assembly error calculation in Jacobian-torsor model. A three-part assembly is taken as an example to illustrate the calculative process.

2. Influence factors of part deformation

2.1. Temperature

Most materials change length as temperature is changed. As a result of this change, the dimensions and tolerances of a product become at variance with the design values. Therefore, thermal impact must be taken into effect during the design process, particularly when a complicated product with multiple components and various materials operates under a wide range of temperature. As shown in Fig. 1, a hollow cylinder in working condition produces deformation because of the induced heat.

2.2. Gravity

The gravity leads to the deformation of the component, as shown in Fig.2. In an ideal 3-dimension model, part locates in desired position, the centre of the shaft is generally coincide with hole centre in a hole shaft assembly. However, in actual situation, part will shift from its axis under the influence of gravity, offset distance is associated with the diameter of the hole and shaft, and their geometric tolerance.

The amount of deformation is proportional to the mass of parts, and can be calculated using finite element analysis. Then it is incorporated in the tolerance stack up progress, improving the precision of the error estimation.

Fig. 1 Heat deformation nephogram. (a) Overall deformation; (b) Axial deformation.

Ideal condition

Under gravity

gravity

Fig.2 The actual parts location under the action of gravity

2.3. Load

During a practical working situation, the force field applied to single parts or assembly results in geometric errors and dimensional changes in size, direction and magnitude in spatial six degrees of freedom. Taking a bourdon tube as example, it bears bending load when it works, and the whole bourdon tube tilts to the side of the load. The larger the bending load, the bigger the bending deflection. Concrete displacement can be obtained by finite element analysis.

Fig.3 Bourdon tube bears bending load. (a) Bourdon tube model; (b) Load diagram; (c) Strain contours.

3. Assembly error calculation considering deformation based on Jacobian-torsor

3.1. The Jacobian-torsor model

In the assembly process, the final product quality is determined by stack up, coupling and propagation of all errors. The unified Jacobian-Torsor model introduced by Desrochers et al [25] is an innovative tolerance analysis method which uses the small displacement torsor (SDT) [26] for tolerance representation and the Jacobian [27] for tolerance propagation, The unified Jacobian-torsor model is built that combine the advantages of these two methods.

The small displacement torsor can directly represent potential variations along and about all three Cartesian axes in its generic form. Jacobian matrix has been proposed to map all SDT from their local origin to the point of interest. It is a matrix whose columns are extracted from the various homogeneous transform matrices relating the reference frames of the functional elements (FE) to that of the functional requirement (FR). Therefore, Jacobian-torsor model can be expressed as follows:

[FR] = [J][FE] (1)

where the functional requirement (play, gap, clearance) represented by [FR] is the translational and rotational vectors of general assembly. [FE] is related to the functional elements for assembly parts mathematically. [J] is a Jacobian matrix expressing a geometrical relation between a [FR] vector and corresponding [FE] vector; Spreading eq. (1) into:

--UWMU-WJr'MUl

[a â]

[[i j]_ »

[a â] [/? /?]

L F_ F\ }[i i]

where [/1/2/3/14/5/6^1 is the Jacobian matrix which describes the relationship between vector [FR] and [FE]. i represents the number of functional elements on the

tolerance transport chain.

UVW Ç£ P S '■ lower limits of the tolerance intervals in six freedom.

u v w a fi S ■ upper limits of the tolerance intervals in six freedom.

The Jacobian matrix is formulated by:

]3,3 ... W]3,3([Rô]3*3.M3,3)"

[ J U = [ J I'o =

[R0 ] 3X3 ■ [RPT ] 3,

is a skew-symmetric

where^R^ = [ClC^C3 ]is the local orientation changes

of frame i with respect to original frame 0.

[RPTi ] = [C1C2C3 ] , C1C2C3 represents the direction

change of tolerance with respect to the three coordinate axes of frame i.

0 —dzi dyi [wn = dzi 0 -dxi

—dyi dxi 0 matrix, indicates the position changes on frame n with respect to frame i, in which dxn = dx — dx. ;

1 in l

dyi = dyn — dyi; dzn = dzn — dzt. The final [FR] can be calculated based on above equations.

3.2. Assembly error calculation taking into account part deformation

The factors referred in Section 2 may changes the dimension, shape or position of parts, these errors will be added up and magnified in the assembly process. Deformation in parts is shown as a translational vectors in the coordinate system and rotational vectors rounding the coordinate system, covered as follows:

S = [Am Av Aw Aa Afl Ay ]T (4)

where A u, Av and Aw are respectively three translational vectors of the origin of the coordinate system on axes x, y, and z.Likewise, A a, Aft and Ay are rotational vectors around the axes.

The specific value in vector S can be obtained with the combination of ANSYS and MATLAB. Finite element is used to get the coordinate information for discrete points in characteristics surfaces while MATLAB can associate corresponding surfaces and extract real derived element, to gain its position and direction information. The process for obtaining integral deformation information is shown in Fig. 4.

Fig.4 Steps for deformation information obtainment

Stepl: Calculate ideal derived feature displacement information including position and direction vector. Step2: Importing three-dimensional assemblage diagram into the finite element analysis software, by meshing, adding constraint and imposing temperature, gravity and load, then the deformation for each node is obtained. Step3: Using GPS theory to gained nodes, associating deprived feature.

Step4: Comparing the vectors acquired in Stepl and Step3 to get translational vector Av Aw] and rotational vector

[AaAfi Ay ].

Part deformation brings about small change in coordinate system which can be calculated in above four steps. In consequence, functional element FE varies. Variables in Eq. (3) will be updated as:

R ii= R OLx ][ Cy, ~\[czi ]

[c„r[ Ci J1 [Ci J-1

0 ~dzi dyi dz*, 0 —dxf, -dyi dxi 0

where |C ],^c .J and[C ] are transition matrix which

rotates A a, Aft and Ay respectively around the corresponding axis on frame i. They can be computed as following.

[ C„ ] =

[Ci ] =

0 cos Aa —sin Aa 0 sin Aa cos Aa cos A/? 0 sin AjB 0 1 0 — sin A/? 0 cos Ap cos Ay — sin Ay 0 sin Ay cos Ay 0 0 0 1

dxi = dxn, — dxv — (dx" + Aun ) — (dxi + Aui)

= dxi + ( A un - A ui) dyi = dyi + ( Au^ - Aui ) ; dzi = dzi + (Awn - Awi) Combine Eq. (4-7), we can get:

M«[RL ... K' L(MJM

[03,3 ••• MJ^L

Besides, [FE] can rewrite from equation following. [FE]m,=[[T] (9)

In consequence, [FR]' affected by multiple fields is accessible by uniting Eqs.8-9. Assembly error calculation process after deformation is shown in Fig. 5.

Parts deformation

Translator Vector

[Aw Av AW]

Rotational Vector

[AaApAy ]

Fig. 5 Assembly error calculation

Parameters in Fig. 5 can be gotten by Eqs.5-9, then Jacobian-torsor is modified for deformed assembly accumulated error calculation. For contrast purposes, the error obtained considering parts deformation is compared to the case not influenced by multiple fields. Therefore deformation influence can be clearly observed.

4. Case study

A simple assembly consist of three parts is taken as an example in this case study. Its shape, dimension and tolerance are shown in Figs. 6 and 7. The range of fluctuation for apex in part 3 is the key issue to ensure the position precision because it is always used to localization-clamp.

part 3

part 2

Fig.6 Assembly model

As a result, dx", dyf and dzj1 update into:

Fig.7 2-dimension model for assembly

- ► Internal pair ~~Contact pair

(5>-Part3<4>

Fig. 8 Connection graph

Based on the connected relations among the assembly, the connection graph is described in Fig.8. Then Jacobian-torsor equation of the assembly without consideration of deformation can be established as follows: FR=FR1+FR2=[[J]i[J]2[J]3[J]4][[FE]I [FE]2 [FE]3 [FE]4] -0.4000 0.4000" -0.3522 0.3522 -1.6102 1.6102 -0.0020 0.0020 -0.0079 0.0079 -0.0054 0.0054

FR is the final output errors (functional requirements) of the error propagation chains.

In view of part deformation, under the multiple influence of temperature field, gravity field and load field, mainly distorted parts are part2 and part3. GPS theory is applied to gain real derived feature, then variation information is approached to modify Eq. (1).

Fig.9 Principal deformed portion.

Fig. 10Contour fitting of the deformation section.

As shown above, Fig.9 (a) and (b) indicates the principal deformation portion, that is, respectively part3 and its matching part2. Fig. 9 displays contour fitting of the deformation section. Real derived feature association results show that the direction vector of Fig. 10 (a) and (b) are separately (1,-2.9506 X 10"4,1.2681 X 10"3 ) and (1,0.0013, 8.1162 X 10"3) compared to original vector (1,0,0) and (1, 0, 0).

According to Section 3.2, parts deformation results in the changes of Jacobian matrix [J] and torsor matrix [FE], FR' = TJ1 T FE 1

L JFEi' L JFEi'

-0.3991 0.3991" -0.3531 0.3531 -1.6111 1.6111 -0.0021 0.0021 -0.0079 0.0079 -0.0054 0.0054

Comparing the result that consider deformation with the value of FR which ignores deformation, assembly error range increases in the y-direction and z-direction while value in x-direction decreases, that is, directly affects the approachable distribution of apex in final part. Therefore, localization-clamp precision can't be ensured.

5. Conclusions

This study proposes an accumulated error calculation method for assembly which considers the deformation resulting from the temperature, self-weight and load in actual working condition. Jacobian-torsor model is used to compute the integrating error in the whole error propagation chains for assembly. Case result proves that part deformation has make difference to assembly error calculation. Therefore, taking deformation error into consideration in design stage can optimize product performance in advance.

This article only obtains the calculation error results in extreme dimension, unapproachable to get specific error value. Influence factors of deformation are not limited to the referred three. The further exploration will be studied in future research.

Ackn owledgments

This research was supported by the National Nature Science Foundation of China (No. 51275464 and 51575484) and Science Fund for Creative Research Groups of National Natural Science Foundation of China (No.51221004).

6. Reference

[1] Bensheng Xu, Meifa Huang, Yanru Zhong. Research on robotic kinematics-based modelling of accumulation of three-dimensional tolerance [J]. Machinary Design & Manufacture, 2007, 01:105-107.

[2] Xiaoqing Tang, B. J. DAVIES. Computer aided dimensional planning [J]. International Journal of Production Research, 1988, 26:283-297.

[3] Paul J. Faerber. Tolerance analysis of assemblies using kinematically derived sensitivities [D]. Department of Mechanical Engineering Brigham Young University.1999.

[4] C. Lu, J. Y. H. Fuh, Y. S. Wong. Evaluation of product assemblability in different assembly sequences using the tolerancing approach [J]. International Journal of Production Research, 2006, 44(23):5037-5063.

[5] Laperrière L, EIMaraghy HA. Tolerance analysis and synthesis using Jacobian transforms. CIRP Ann—Manuf Technol 2000; 49(1):359-62.

[6] Davidson J K, Shah J J. A new mathematical model for geometric tolerances as applied to round faces [J]. Journal of Mechanical Design, 2002, 124(4):609-622.

[7] Giordano, M., Samper, S., and Petit, J.-P., 2007, "Tolerance Analysis and Synthesis by Means of Deviation Domains, Axi-Symmetric Cases," Models for Computer Aided Tolerancing in Design and Manufacturing, Springer, Netherlands, pp. 85-94.

[8] Chase K W, Gao J, Magleby S P, et al. Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assemblies [J]. Iie Transactions, 1996, 28(10):795--807.

[9] Desrochers, A. Modeling three dimensional tolerance zones using screw parameters. In: Proceedings of the DETC 25th Design Automation Conference, 12-15 September, 1999, Las Vegas, Nevada, USA.

[10] Desrochers A, Rivière A. A matrix approach to the representation of tolerance zones and clearances [J]. International Journal of Advanced Manufacturing Technology, 1997, 13(9):630-636.

[11] Jingsong Gao, Kenneth W. Chase, Spencer P. Magleby. Generalized 3-D tolerance analysis of mechanical assemblies with

small kinematic adjustments [J]. Iie Transactions, 1998, 30(4):367-377.

[12] Ghie W. Statistical tolerance analysis using the unified Jacobian-Torsor model [J]. International Journal of Production Research, 2010, 48(15):4609-4630.

[13] Samper S, Giordano M. Taking into account elastic displacements in 3d tolerancing: Models and application [J]. Journal of Materials Processing Technology, 1998, 78(1): 156-162.

[14] Pierre L, Teissandier D, Nadeau J P. Integration of thermomechanical strains into tolerancing analysis [J]. International Journal on Interactive Design & Manufacturing, 2009, 3(4):247-263.

[15] Jayaprakash G, Thilak M, Sivakumar K. Optimal tolerance design for mechanical assembly considering thermal impact [J]. International Journal of Advanced Manufacturing Technology, 2014, 73(5-8):859-873.

[16] Govindarajalu J, Karuppan S, Manoharan T. Tolerance design of mechanical assembly using NSGA II and finite element analysis[J]. Journal of Mechanical Science & Technology, 2012, 26(10):3261-3268.

[17] Benichou, S. and B. Anselmetti. 2011. Thermal dilatation in functional tolerancing. Mechanism and Machine Theory, p. 15751587.

[18] Baker M and Steinrock G. Dimensional and variation analysis. In: ADCATS conference, Provo, UT, 15-16 June 2000.

[19] Armillotta A, Semeraro Q. Critical operating conditions for assemblies with parameter-dependent dimensions [J]. Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture, 2013, 227(5):735-744.

[20] Guo J, Hong J, Yang Z, et al. A Tolerance Analysis Method for Rotating Machinery [J]. Procedia Cirp, 2013, 10:77-83.

[21] Walter M S J, Spruegel T C, Wartzack S. Least Cost Tolerance Allocation for Systems with Time-variant Deviations [J]. Procedia Cirp, 2015, 27:1-9.

[22] Barari A. Tolerance Allocation Based on the Minimum Deformation Zone of Finite Element Structural Frame Analysis [J]. Computer-Aided Design, 2013, 10(4):629-641.

[23] Weimin Zhang, Can Chen, et al. Tolerance modeling in actual working condition based on Jacobian-Torsor theory [J]. Computer

[24] Jianyong liu. Calculation method for assembly error with consideration of part deformation [J]. Computer integrated manufacturing system, 2015, 21(1):94-100.

[25] Desrochers A, Laperriere L, Ghie W. Application of a Unified Jacobian—Torsor Model for Tolerance Analysis [J]. Journal of Computing & Information Science in Engineering, 2003, 3(1):2-14.

[26] Bourdet P, Clement A. Controlling a complex surface with a 3 axis Measuring Machine. Annals of the CIRP, Vol. 25, Manufacturing Technology, pp. 359-361, 1976, Janv.

[27] Laperriere L, Lafond P. Modeling tolerance and dispersions of mechanical assemblies using virtual joints. In: CD-ROM proceedings of ASME 25th design automation conference. Fukuoka, Japan, 1999, pp 330-336