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Statistical Variation Analysis Using Pearson Distribution Family Based on Jacobian-Torsor Model

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Statistical Variation Analysis Using Pearson Distribution Family Based on Jacobian-Torsor Model MATEC Web of Conferences 139, 00011 (2017) DOI: 10.1051/matecconf/201713900011 ICMITE 2017 Statistical Variation Analysis Using Pearson Distribution Family Based on Jacobian-Torsor Model Siyi Ding1,2, Sun Jin1,2, *, Zhimin Li1,2, Fuyong Yang1, Jia Lin2 1State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China 2Shanghai Key Laboratory of Digital Manufacture for Thin-walled Structures, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China Abstract. Assembly variations are unavoidable due to parts’ geometrical errors. Statistical variation analysis is an effective method to quantitatively predict product quality in the original design stage. However, traditional methods can't handle the problem of abnormal distribution of the actual variation variables. Meanwhile, they are underdeveloped in regard to the complex geometrical errors in spatial 3D state. To overcome this problem, firstly, Jacobian-Torsor model is used to build the variation propagation, which is well suited to a complex assembly that contains large numbers of joints and geometric tolerances; secondly, Pearson distribution family is adopted to determine probability distribution pattern and build probability density function. By comparing results of the suggested method to the Monte Carlo method, it is observed that this novel method has the same accuracy, but much higher efficiency. The results also demonstrate that probability distribution types of the parts variations have a significant impact on the final assembling variation. 1 Introduction into a core of some commercial softwares such as CETOL, VSA, 3DCS, etc. A big disadvantage to this Variation is inherent during any manufacturing and approach, however, is that large amounts of data samples assembly process owing to the variability of the part, are required to achieve accurate predictions [7], which datum, positioner, fixture, etc. They cause minor inevitably result in time-consuming computation. Thus variations in components from the nominal geometry [1]. there is a clear need to adopt a simple and rapid Statistical variation analysis is an effective way to probabilistic approach for statistical variation analysis. predict product quality quantitatively in the design stage Yang et al. [8] studied the probability distribution of [2]. Traditional methods are still based on engineering table-axis error in aero-engine assembly. Probability experience, extremum method [3, 4], or the mean square density functions (PDF) were derived based on Rayleigh root method [5, 6]. The results obtained by extremum distribution. Ghie [9] applied uncertain measurements method are overly pessimistic which will cause high into Jacobian-Torsor model. But the PDF deduced could fabricating cost, while the mean square root method only not consider skewed distribution of variables. gives results for the mean-square variation which is Ahsanullah and Hamedani [10] attempted to investigate primarily used to solve the plane size-chain problem. various characterizations of certain univariate continuous These methods are underdeveloped in regard to the distributions, providing a base of further research of geometrical errors in spatial 3D state. Meanwhile, they multivariable continuous distributions. Kharoufehy and widely adopt normality assumption which is convenient Chandraz [11] adopted a multivariate kernel density for statistical evaluation. estimate to approximate the probability distribution. But As is known to all, most variables of parts variations it was difficult to determine the smoothing parameter (h). disobey normal distribution in the actual machining In this paper, we will introduce an innovation of Pearson process. For instance, if tool wearing plays a leading role distribution family [12] to model PDF that covers in the machining process, the actual variation variables polymorphic probability distributions. will satisfy uniform distribution; if one manufacturing In addition, Jacobian-Torsor theory [13] will be used process is accomplished using two machine tools, the to model the variation propagation, which is well suited mixing variables of parts variations will follow bimodal to a complex assembly that contains large numbers of distribution; when turning shaft journals or apertures, joints and geometric tolerances. It combines the operators usually keep a certain allowance, and this will advantages of the torsor model which is suitable for lead to skewed distribution. Monte Carlo (MC) tolerance representation and the Jacobian matrix which simulation is suitable for the data of abnormal is suitable for tolerance propagation. distributions. Currently, MC method has been developed * Corresponding author: [email protected] © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 139, 00011 (2017) DOI: 10.1051/matecconf/201713900011 ICMITE 2017 2 Related works types of distributions. Pearson system adopts four parameters to characterize each distribution. The 2.1 Jacobian–Torsor model probability density function (PDF) f(x) depends on the following differential equation: The unified Jacobian-Torsor model is composed of the df() x (a ax ) f () x 01 2 (4) Jacobian matrix and torsor model. Torsor model uses dx b01 b x b 2 x three translational vectors and three rotational vectors to While a1=1, the parameters a0, b0, b1, and b2 depend express the position and orientation of an ideal surface. entirely on the first four order central moments of Supposing that S0 is the nominal position and S1 is an distributions μ1, μ2, μ3 and μ4. It can also be expressed as arbitrary variational surface relative to S0 with tolerance t, 2 3 2 moment ratio, i.e. β1=μ3 /μ2 and β2=μ4/μ2 , as follows: the torsor model of S which represents position and 1 ab (5) orientation can be represented as: 01 22 2 2 T bA0(4 43 3 )B (4 21 3 ) (6) T10 uvw (1) bA( 3 4 ) B ( 3) where u, v, w are three specific vectors along the axes x, 1 34 12 (7) 2 26 y and z in local reference system respectively; likewise, bA2(2 43 3 6 ) B (2 21 3 6) (8) α, β and γ indicate the vectors around the axes x, y and z where A (102 18 6 12 21 ) (9) respectively. 43 B (10 12 18)1 These vectors are transferred to the objective 21 (10) controlled variable by virtue of the Jacobian matrix. The general solution of f(x) are: Jacobian matrixes describe relative positions, as well as a ax f( x ) C exp 01 dx (11) relative orientations between local reference frames and 2 b01 bx+ bx 2 the global reference frame. The unified Jacobian-Torsor To be specific, Pearson system mainly contains seven model can be expressed as: T distribution patterns, which are elaborated in [14]. These uu, uu,, uu types can be distinguished according to the different vv, vv,, vv distributions of roots of the quadratic equation 2 2 2 b0+b1x+b2x =0. Supposing D=b1 -4b0b2, λ=b1 ÷(4b0b2), ww, ww,, ww the Pearson distribution pattern can be judged by the JJFE1 FEn (2) , ,, value of D and λ, as provided in Table.1. , ,, Table 1. Pearson distribution types. , ,, FR FE1 FEn a1 D λ Distribution pattern Remarks where α, β, γ, u, v and w are the same as Eq. (1); n is the 0 0 0 Exponential distribution ∞ number of functional elements (FEs); , is tolerance - Pattern VII b0=0 (-∞,0) Pattern I interval where α should belong to; likewise, other torsors ＞0 0 Pattern II of components follow the similar way as α. (1,∞) Pattern VI ≠0 There are two types of FE pairs in assembly, i.e., ∞ Pattern III internal FE (IFE) pairs which are composed of two FEs 1 Pattern V =0 b =b =0 on the same part and contact FE (CFE) pairs which are Uncertain Normal distribution 1 2 made up of two FEs on different parts. Variations of ＜0 (0,1) Pattern IV each FE pair can be transferred to functional requirement (FR) with a Jacobian matrix, which of the ith FE can be expressed as: 3 Statistical analysis RRi Wni () RR 0033Pti 33i 33 33 Pti 33 According to the Pearson system’s characteristics, J (3) distribution pattern can be determined and PDF can be FEi 0 RRi derived while the first four central moments are known. 330 33 Pti 33 i th As for arbitrary distributions, the first-order central where [R0 ] represents the local orientation of the i moment μ1 is identically equal zero; the second-order frame with respect to the 0th frame which is global central moment μ2 reflects data dispersion degree, which reference system; [RPti] is a projection matrix designating 2 is equal to variance σ ; the third-order central moment μ3 the unit vectors along local axes respectively for 3 is related to coefficient of skewness S, and S=μ3/σ ; and tolerance zone tilted according to the direction of 4 n the μ4 is related to coefficient of kurtosis K, and K=μ4/σ . tolerance analysis; [Wi ] is a skew-symmetric matrix [13] Due to the average value, variance, skewness and allowing the representation of the vector among the ith th kurtosis of a probability distribution can visually reflect and n frame (end point). the shape of the distribution, we will solve these four parameters in this paper to indirectly gain the first four 2.2 Pearson distribution family central moments. Eq. (2) can be represented as yFR=f(ui, vi, wi, αi, βi, Pearson system of distributions was firstly proposed by γi,), (i=1,2,…,n), where yFR is the FR’s variation, and (ui, statistician K. Pearson in 1895, which included various 2 MATEC Web of Conferences 139, 00011 (2017) DOI: 10.1051/matecconf/201713900011 ICMITE 2017 vi, wi, αi, βi, γi,) is the FE’s shifting variation or rotating Figure. 2. Four-stage assembly: (a) parts drawing, (b) variation in all directions.
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