Geometric Tolerance Simulation Model for

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among various tolerances and Rule#1 etc. (ASME Y14.5M). In conventional tolerancing, tolerances are assigned to dimensions to specify allowable variation ...
TRANSACTIONS OF NAMRI/SME 2010

Transactions of NAMRI/SME, Vol. 38, 2010, pp. 363-370 TP10PUB57

Geometric Tolerance Simulation Model for Rectangular and Circular Planar Features authors WENZHEN HUANG and BHARATH REDDY KONDA Department of Mechanical Engineering University of Massachusetts-Dartmouth North Dartmouth, MA ZHENYU KONG School of Industrial Engineering & Management Oklahoma State University Stillwater, OK

abstract Geometric tolerance modeling and simulation methods are developed for rectangular and circular planar features. The method decomposes the feature variation as translational and orientation components and proposes a sequential simulation technique by using conditional probability to address the interaction between size and orientation tolerances. Numerical case study is conducted to demonstrate and validate the proposed method.

terms Planar feature, GD&T, Tolerance analysis, Tolerance design, Monte Carlo simulation

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GEOMETRIC TOLERANCE SIMULATION MODEL FOR RECTANGULAR AND CIRCULAR PLANAR FEATURES

Wenzhen Huang and Bharath Reddy Konda Department of Mechanical Engineering University of Massachusetts-Dartmouth North Dartmouth, MA Zhenyu Kong School of Industrial Engineering & Management Oklahoma State University Stillwater, OK

KEYWORDS Planar feature, GD&T, Tolerance analysis, Tolerance design, Monte Carlo simulation

ABSTRACT Geometric tolerance modeling and simulation methods are developed for rectangular and circular planar features. The method decomposes the feature variation as translational and orientation components and proposes a sequential simulation technique by using conditional probability to address the interaction between size and orientation tolerances. Numerical case study is conducted to demonstrate and validate the proposed method.

design and manufacturing by defining allowable variations on part geometric features. Thus, it is imperative to create a tolerance model that is capable of simulating random variations and incorporating them into product function models for probabilistic design in the face of manufacturing uncertainties. In statistical tolerance design the first step is to replicate part mating features subject to GD&T specifications. Then, the functionality and performance of a product can be predicted through Monte Carlo simulation. The simulated features must fully conform to GD&T standards. All the interrelated requirements among individual tolerance zones as per GD&T standards must be fully satisfied, such as containment relations that allow floating/tradeoff among various tolerances and Rule#1 etc. (ASME Y14.5M).

INTRODUCTION Tolerancing is of great importance in design and manufacturing. It is almost impossible to fabricate perfect part features because of uncertainties in manufacturing. Probabilistic design that accounts manufacturing induced variability is vital ensure desirable product quality/reliability. inputs to probabilistic design, tolerances link

for to As the

In conventional tolerancing, tolerances are assigned to dimensions to specify allowable variation intervals in the directions of the dimensions. Independency is assumed and their statistical models are relatively simple, defined by univariate distributions. The dimensional tolerance model has been applied to variation analysis of multistage assembly systems (Ceglarek et al. 2004, Huang et al. 2007). Geometric tolerancing assigns values to certain

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attributes of a feature (e.g. location, orientation, and form etc.). However, complex interactions (containment, floating in composite tolerance etc.) among these tolerances make statistical tolerance modeling difficult. Requicha pioneered the earliest development of offset zone theory for tolerance modeling (Requicha, 1983). The Minkowski sum is proposed to construct tolerance zones which contain all possible variational classes allowed by the tolerances (Jayaraman et al. 1989). Parametric Space Models, which map the variational class into parametric spaces, are widely used in current computer aided tolerancing (CAT) software (e.g. 3DCS, EDS/VisVSA etc.). They use either CAD parametric model in features definition, or abstracted parametric models for tolerance design. The former includes variational surface models (Roy et al. 1998, Turner et al. 1990) that describe variations by changing models coefficients (e.g. polynomial, Bezier, etc). They can be easily embedded into the existing CAD systems. Monte Carlo simulation for statistical analysis can be achieved by assigning distributions to the model parameters. However, the analyses rely largely on the subjective model assumptions or try-and-error method to determine distributions (Whitney et al. 1994). The abstracted parametric models create separate parametric models from the CAD geometric models and are popular in CAT software. A straightforward parameterization is the directly use of discretized points (3DCS, eM-TolMate, EDS/VisVSA). A special type of Parametric Space Model maps variational class into a parametric space which describes the kinematics degrees of freedom of a toleranced feature within the tolerance zone (Whitney et al. 1994). The screws or torsors are used to represent kinematics displacements. T-Map models take into account tolerance zone interactions (Rule #1, floating zones, etc.) (Mujezinovic et al. 2004, Shah et al. 2007). All the kinematics displacements of a feature with perfect form are mapped to points in a hypothetical Euclidean space. T-map can effectively avoid model complexity by ignoring form error, providing a simple applicable solution. In spite of the efforts and advancements made in the last three decades a statistical tolerance modeling method that accounts for interactions among different GD&T requirements and tolerances is not available.

TOLERANCE MODEL FOR PLANAR FEATURES In this section, we introduce planar feature tolerance modeling for statistical GD&T tolerance analysis. We confine the scope of this paper to the rectangular planar and circular planar features subject to regular or composite GD&T requirements. In the following discussion we assume the form variation can be ignored. The tolerances in consideration only include size and orientation. The methods for all types of GD&T tolerances will be considered in future work.

Rectangular Planar Surface A rectangular planar feature freely floats in a size tolerance zone as shown below in FIG. 1. The deviation δ of any point P(x, y) on the feature can be expressed by δ = γ1 + γ2x + γ3y + ε

(1)

where γ1 is measured at the center of the feature, representing translational deviation of the feature along the normal direction of the nominal feature. γ2 and γ3 are shown in FIG. 1, representing orientation errors of the planar about x and y z B

y P

Z3

γ3

Z4 x

Z2

Z1

γ2 A

A FIGURE 1. A RECTANGULAR PLANAR FEATURE IN TOLERANCE ZONE Ts.

axes. ε is composed of form variation components and ignored here. It will be considered with orthogonal expansion in future as in (Huang et al. 2008). By ignoring ε a feature can be characterized by a point (γ1, γ2, γ3). If one can randomly generate a large number of such sample points that conform to GD&T standards, assembly analysis and tolerance design can be conducted. The idea behind Eq.(1) is to express the variation in rigid and deformation modes. The

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first term (mode) represents translational mode in feature’s deviation; the second and third are rotational movements or modes in two directions. ε will be expressed by a linear combination of higher modes. These higher modes may represent warp, twisting, bending, waviness, roughness components, etc. The size and orientation tolerances are specified as Ts and To, i.e. |γ1| ≤ Ts and |γ2x + γ3y| ≤ To. In composite tolerance case the To must be imbedded and floating in Ts. When γ1 approaches the upper boundary of Ts Maximum Material Condition (MMC) requires the reduction of To to ensure a complete coverage of To by Ts. Thus, a viable orientation zone Tv must be the function of random variable γ1: Tv=f(γ1, To) or the orientation tolerance is conditional to the realization of γ1. Specified To, it yields:

We further assume γ2 and γ3 are normally, independently distributed variables. It can be justified by the fact that the orientation variations are largely dominated by fixture positioning errors in two orthogonal directions and machining errors along paths of cutting tools in machining. These factors (causes) are independent. There are also a large number of small random contributors such as vibration, heterogeneity of material, thermo effects etc., lending the distribution approximately normal. Thus, the binormal distribution is expressed by γ3 Tv/B

-Tv/A

γ2 ΩTv Tv/A

T − To ⎧ To if γ 1 ≤ s ⎪⎪ 2 Tv = ⎨ ⎪2( Ts − γ ) Otherwise 1 ⎪⎩ 2

-Tv/B

(2)

It can be seen that with the size (location) deviation 1 approaching the tolerance boundary the orientation zone approaches zero. This ensures the MMC being satisfied. For any given Tv the deviation caused by orientation variation must be |γ2x + γ3y| ≤ Tv. It is easy to show that |γ2x + γ3y| ≤ Tv holds if |γ2A + γ3B|≤Tv, -γ2A + γ3B≤Tv, and γ2A - γ3B≤Tv

FIGURE 3. TOLERANCE ZONE (RHOMBUS ΩTv) AND (1-α)×100% PROBABILITY REGIONS (ELLIPSES).

f (γ 2 , γ 3 ) =

where T



(-1)

1 ( 2π ) Σ

1/ 2

⎡γ ⎤ ⎡σ γ = ⎢ 2 ⎥ ,Σ = ⎢ 2 ⎣γ 3 ⎦ ⎣0

e− γ

T

Σ −1 γ

/2 (4)

0⎤ σ 3 ⎥⎦ .

Notice that

2

 = c gives a contour of constant density

2 2 of the binormal distribution. Using c = χ2 ( α ) 2

(3) are satisfied. Tv=f(γ1, To)

Ts Ts-To

0

γ1

FIGURE 2. SIZE AND VARIABLE ORIENTATION TOLERANCE ZONES.

These conditions form a parallelogram (rhombus). The size of the rhombus is controlled by the deviation γ1. FIG. 3 shows one such rhombus.

(Johnson et al. 1998), where χ2 ( α ) is the upper (100α)th percentile of a chi-square distribution with 2 degrees of freedom, leads to contours that have (1 – α)×100% of the probability. FIG. 3 shows two contours with α = 0.005 or 99.5% probability for the  falls into the ellipses. The larger ellipse passes four vertices of the rhombus, whereas the smaller one is imbedded in the rhombus, contacting the four sides of it with four points. Thus, the larger contour covers a region of 2

2

⎛ γ3 ⎞ ⎛ γ2 ⎞ 2 ⎜⎝ σ ⎟⎠ + ⎜⎝ σ ⎟⎠ ≤ χ2 (α = 0.005) = 10.6 2 3

(5)

This leads to

σ2 =

Tv Tv and σ 3 = 3.256 A 3.256B

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(6)

To determine the smaller ellipse, let the following equations have precisely one solution (contacting point):

⎧⎛ ⎞ ⎛ ⎞ γ3 γ2 ⎪⎜ +⎜ 2 2 ⎟ ⎟ =1 ⎨⎝ σ 2 χ2 (α = 0.005) ⎠ ⎝ σ 3 χ2 (α = 0.005) ⎠ ⎪ γ 2 A + γ 3 B = Tv ⎩ 2

and y axes. The deviation of any point P(x, y) on the circle of the feature is given by: δ = γ1 + γ2x + γ3y = γ1 + γ2Rcosθ+ γ3Rsinθ

2

It yields

Tv Tv and σ 3 = σ2 = 4.604 A 4.604B

(8)

where R, θ are shown in FIG. 5. The extreme points must fall within the size and the orientation tolerance zones, denoted by Ts and To, respectively. This requires |δ = γ1 + γ2x + γ3y|∈ΩTs |γ2x + γ3y|∈ΩTv

(7)

z Eqs. (6), (7) and binormal function (4) can provide a method to simulate orientation variation. Since the orientation tolerance range Tv relies on 1’s value (Eq. (2)), a procedure for simulation that generates 1, 2, and 3 sequentially can easily account for this tolerance containment constraint. It can be found in FIG. 3 that there are four regions in the larger ellipse falling outside the tolerance region (rhombus). This implies that the larger ellipse (Eq. (6)) gives a liberal result that leads to more non-conforming samples than expected in simulation. The smaller ellipse, on the other hand, is fully covered by the orientation region with four regions in the rhombus but outside the ellipse. This fact implies the Eq. (7) derived from the internally imbedded ellipse will provide a more conservative result. The accurate conformity estimation in region ΩTv of above two cases can only be γ1 obtained through Monte Carlo simulation. The above derivations ΩT emphasize the γ3 orientation tolerance γ2 subject to the variation of size deviation γ1. The three dimension tolerance region ΩT FIGURE 4. 3-D forms space shown in TOLERANCE REGION. FIG. 4.

A circular planar surface floats in a tolerance zone as shown in FIG. 5. The similar expression as Eq.(1) is used to characterize the feature variations. γ2 and γ3 in Eq. (1) represent orientation errors of the planar surface about x

(10)

R y

Ts

θ x

L-Ts/2

A FIGURE 5. A CIRCULAR PLANAR FEATURE WITHIN A TOLERANCE ZONE.

Similarly, To must fall into Ts and float in it. This relationship yields Tv=f(γ1, To):

T − To ⎧ To if γ 1 ≤ s ⎪⎪ 2 Tv = ⎨ ⎪2( Ts − γ ) Otherwise 1 ⎪⎩ 2

(11)

The extreme points on the feature can be derived by letting

∂δ = 0. It yields ∂θ

δ extreme = γ 1 + R γ 22 + γ 32

(12)

It can be assumed that the extreme point is uniformly distributed along the circle, or θ ~ U(0, 2π)

Circular Planar Surface

(9)

(13)

U(0, 2π) is a uniform distribution in [0, 2π]. Thus, the two orientation angles γ2, γ3 about x and y can be combined into a single one:

γ ext = γ 22 + γ 32

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(14)

ext indicates the orientation angle between the feature surface and nominal feature surface. Thus the orientation tolerance constraint Eq. (10) is simplified to:

Rγ ext ≤ Tv

(15)

The tolerance region ΩT is visualized in FIG. 6. From above analysis the circular feature can be simulated by using the following models: γ1 ~ N(0, Ts/C1)

(16)

γext=N(0, Tv/(C2R))

(17)

By using these two models and letting C1 = 6, C2=3, the marginal conformities of |γ1| ≤ Ts and γ2 Ts-To To/R -Ts/2

ΩT

tolerance analysis one needs to generate samples that must conform to tolerance requirements with specified probability. Monte Carlo technique provides a powerful tool for the sample generation provided that the joint distribution of (γ1, γ2, γ3) is given. As abovementioned the distribution of orientation variation f(2, 3) relies on the realization of γ1, through Eqs. (2)~(7). Therefore, it is difficult, if not impossible, to derive the joint distribution f(1, 2, 3) from the specified tolerances and their complex relationships. A strategy is proposed below to address this problem. In dimensional tolerance analysis the distribution of a dimension is determined by assigning σ = T/6 (T is the specified bilateral tolerance range). p = 99.73% tolerance conforming probability can thus be ensured under normality assumption. For geometric tolerance analysis the generated samples should satisfy p= P[(γ1, γ2, γ3)∈ΩT]

Ts/2 γ1

-To/R FIGURE 6. TOLERANCE REGION OF CIRCULAR PLANAR FEATURE.

|γext| ≤ Tv will be 99.73% and total conformity will 2 = 99.46%. To ensure a be (99.73%) conventional 99.73% conformity, one can choose C1=6.4 and C2=3.2. These values can be picked up from a normal distribution table. These values lead to marginal conformities of the two tolerances about 99.86% and total conforming probability becomes about 99.73%. Similarly, the tolerance containment constraint between size and orientation tolerances can be treated in the same way as in the earlier section by generating 1, 2, and 3 sequentially in simulation. The planar feature tolerance models proposed in this section avoid direct modeling of joint distribution of size and orientation variations and facilitate a sequential procedure for simulation as presented below. STATISTICAL SIMULATION OF PLANAR FEATURES To replicate a planar feature for statistical

(18)

where p, ΩT, P[ ] are specified conformity, tolerance region (FIGS. 4, 6), and probability, respectively. Defining the events E = [(γ1, γ2, γ3) ∈ ΩT], E1 = [γ1 ∈ ΩTs], and E2 = [(γ2, γ3) ∈ ΩTv], thus, E = E2 ∩ E2. Using multiplication law it yields P(E) = P(E1∩E2) = P(E2|E1)*P(E1) P[(γ1,γ2,γ3)∈ΩT] = P[(γ2,γ3)∈ΩTv|γ1]*P[γ1∈ΩTs] (19) where ΩTs and ΩTv are tolerance zones for size (Ts in FIG. 2) and orientation (FIG. 3). Eq. (19) directly characterizes the interactions between tolerances, thus is different from the joint normal distribution in literature for geometric tolerance modeling (Whitney, 2004). One of the advantages of this method is to avoid direct modeling of joint distribution f(γ1, γ2, γ3). In addition, the separation of event E1 and E2 enables simulating these events sequentially as manufacturing processes proceed. The simulation can be designed to generate event E1 (γ1) first followed by (γ2, γ3) generation. The total conformity p is the product of conformities of the two events, i.e. p = p1*p2. Decomposition of p to the product of p1 and p2 provides the flexibility in tolerance design. Different marginal conformities can be assigned to represent designer’s preference or to differentiate the significances of

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conformity to size and orientation tolerances. When both p1 and p2 are assigned with 0.9973 the total conformity will be p = 0.9946. To achieve the same conformity of 0.9973 the marginal conformities (p1 and p2) should be bigger than 0.9973 (e.g., 0.9987). This can be achieved by choosing constants Ci, as shown below. The simulation procedure proceeds straightforward. We propose the following strategy for the rectangular planar feature simulation: th

Step 1: In i run, generating a random sample:

γ 1i ~ N(0, s1), s1 = Ts/C1. This will allow a specified conformity p1 (e.g. C1 = 6, p1 = 99.73% or C1 = 6.4, p1= 99.86%) for the size tolerance. i

Step 2: Updating Tv = f( γ 1 , To) using Eq. (2). Step 3: Determining binormal distribution for orientation variation: f(g2, g3) ~ N(0, Σ) using Eqs. (4)~(7). The conformity p2 can be determined through the selection of a in chi-square distribution Eq. (5). Step 4: Generating a sample of



i 2

, γ i3

)

from

f(g2, g3). i i i Step 5: Combining γ 1 with γ 2 , γ 3 to provide a

(

i i i i sample P = γ 1 , γ 2 , γ 3

)

representing a

simulated planar feature as described in Eq. (1). Step 6: Repeat Steps1~5 to replicate N samples, i i.e. {P }. Similarly, the procedure for circular planar features simulation can be established as well. th

Step 1: In i run, generating a random sample:

γ 1i ~ N(0, s1), s1 = Ts/C1. This will allow a specified conformity p1 (e.g. C1 = 6, p1 = 99.73%; C1 = 6.4, p1 = 99.86%, etc.) for the size tolerance. i

Step 2: Updating Tv = f( γ 1 , To) using Eq. (11). Step 3: Determining normal distribution for orientation variation: γext = N(0, Tv/(C2R)) (Eq. (17)). The marginal conformity p2 can be determined from normal distribution table using C2 (e.g. C2 = 3, p2 = 99.73%; C2 = 3.2, p2 = 99.86%, etc.).

Step 4: Generating a sample Tv/(C2R)).

γ iext

~ N(0,

i i th Step 5: Combine γ 1 with γ ext to provide the i

sample

(

)

P γ 1i , γ iext . It represents a

replication of planar feature as described in Eq. (1) for the circular planar feature. Step 6: Repeat Steps1~5 to replicate N samples, i i.e. {P }.

SIMULATION AND VERIFICATION Numerical case study was conducted to demonstrate and validate the proposed methods. Monte Carlo simulations were carried out to draw large samples using the proposed tolerance model and simulation procedures. These samples were compared with boundaries of the tolerance zone for conformity checking. The yield in each case was then estimated by calculating the ratio of the number of conforming samples and sample size N. In general tolerance design application, less samples are usually sufficient such as suggested in (Cvetko et al. 1998), i.e. N = 3,000~5,000. However, a large sample size of N = 100,000 was chosen in the case study to ensure the accuracy of the yield calculation. Matlab codes for these procedures were prepared for simulation. The calculation time in the simulation was mainly consumed in conformity checking. This check is only designed for validation purpose and unnecessary in tolerance analysis in application. It is our experience that the simulation with 3000~5000 samples takes almost no time. There are two groups of cases: Case I and Case II form group 1 for rectangular planar feature simulation; the rests are in group 2 for circular planar feature simulation. Cases I and III assigned marginal tolerances with a conventional way, i.e. T/2 = 3σ, leading to the total conformities less than 99.73%. Cases II and IV showed the improvement through the adjustments of the model constants. Simulation Case I: A rectangular planar feature with A = 40, B = 60, size tolerance Ts = 0.5, orientation tolerance To = 0.2. 99.5% probability

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γext

γ3

ΩT γ2

γ1

FIGURE 7. TOLERANCE CONFORMITY ESTIMATION BY SIMULATION FOR A RECTANGULAR PLANAR FEATURE.

FIGURE 8. TOLERANCE CONFORMITY ESTIMATION BY SIMULATION FOR A CIRCULAR PLANAR FEATURE.

contour and orientation tolerance zone ΩTv is shown in FIG. 7. Eq. (7) is used to calculate the standard deviations of 2, 3. N = 100,000 was chosen and the total conformity p = 0.9951 was obtained by yield calculation (yield = 1 – ratio of non-conformity/conformity samples). p1 = 99.73% and p2 = 99.78% (estimated by similar yield analysis method that counts only the nonconformity of region ΩTv).

SUMMARY

Simulation Case II: The same planar feature in Case I was reused. To achieve expected conventional 99.73% total conformity the p1 = 99.86 or C1 = 6.4 was used. N = 100,000 samples were generated. The conformity p = 99.71%. Simulation Case III: A circular planar feature with R = 40, size and orientation tolerances of Ts = 0.5, To = 0.4 was simulated. Both marginal conformities of size and orientation tolerances were assigned as 99.73%, i.e., C1 = C2 = 3. The total conformity of p = 99.45% was obtained. It is very close to its theoretical value of 99.46%. Simulation Case IV: The same planar feature in Case III was reused. To achieve expected conventional 99.73% total conformity the p1 = 99.86 or C1 = 6.4 and C2 = 3.2 were used. N = 100,000 samples were generated. The resultant total conformity p = 99.77%. The sample points in tolerance zone ΩT is shown in FIG. 8. The four numerical case study shows that the proposed models and procedures achieved expected conformity and the interacted tolerance requirements are met automatically. Thus the model and procedure can facilitate automatic tolerance design simulation.

Geometric tolerance models and statistical simulation procedures for rectangular and circular planar features are developed. Based on conditional probability, a sequential simulation approach is developed to account for interaction between size and orientation tolerances such as containment/floating-and-trade-off. It allows simulation of planar features, avoiding direct modeling joint distribution which is very difficult to determine because of the complex interactions among tolerances and rules. The new method decomposes the joint distribution into marginal ones, i.e. size and orientation tolerance distribution models. This leads to a straightforward sequential simulation procedure; and the coding is simple. The model allows assigning different marginal conformity (probability) to different tolerances, accounting for designer’s preference and priorities of tolerances and ensuring total conformity to tolerances. This flexibility can be used to put different priorities to tolerances to meet product function requirements. For example, an engineer can set a higher conformity to orientation than size to closely control the orientation variation. The developed method is the initial step towards the establishment of a comprehensive statistical geometrical tolerance (G/ST) system. The current work decomposes planar feature variation in three rigid modes (one translational and two orientation modes). The future work will include form tolerance (flatness) and models for other geometric features such as cylindrical, spherical, conical etc. The Eq. (1) will be extended to account for more modes by introducing statistical modal analysis (Huang et al. 2002, 2008) and modeling techniques. The

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extended modal model will characterize feature variability in a unified framework, which allows incorporation of GD&T in tolerance design and variation control in design and manufacturing. This effort will extend the variation stackup analysis in assembly systems (Huang et al. 2007) to account for GD&T and compliant components. When incorporated with assembly variation propagation models, it can be applied to statistical variation and tolerance analysis in rigid and compliant assemblies. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of the National Science Foundation (NSFCMMI: #0928609 & #0927557) and UMASSD Chancellor’s Research Grant provided by University of Massachusetts Dartmouth. Appreciation also goes to Dr. Y. Zhou and Dimensional Control Systems (DCS) Inc. for their enthusiastic support and insightful discussions.

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