Tolerance Analysis

Tolerance Analysis

Pat Hammett

Geometric Dimensioning

and Tolerancing (GD&T)

Tolerance Analysis Methods

1

Example: Joining Parts

n

Suppose you want to join the following two parts.

n

They need to be properly aligned anddimensionally

acceptable to insure a good assembly.

Height

weld

Assembly

OK

Poor

Excess

Align

Deviation

weld

weld

NOK

NOK

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Tolerance Analysis

Pat Hammett

What is GD&T?

n

Symbolized notation system to communicate

tolerances through the use ofdatums(references).

n

GD&T is used whenever the location of a part is as

critical or more critical than its size. It insures that

two parts can mate or join properly.

n

Thus, GD&T communicates 2 key issues:

n

Datum Reference System (how part is held).

n

Tolerances for Part Characteristics.

3

Datums

n

Datums define the reference system for a part

to measure or assemble it.

n

reference systems may be absolute (XYZ) or relative (XY).

n

Use pins ~ part holes or slots & clamps ~ part surfaces

Hole:

(4-way locator)

Slot: B2

(2-way locator)

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Tolerance Analysis

Pat Hammett

Symbols and Characteristics

Type of ToleranceCharacteristicSymbol

Individual FeatureFlatness

“Diameter

Individual or

Related Features

Profile

Related FeaturesPosition

“Parallelism

5

Dimension

(2mm total tolerance band relative to datums)

Geometric Tolerance

Square Tolerance

Allows deviation of

for T @45 = +/-1

1mm in any direction

t

X

= 0.7;t

Y

= 0.7

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Tolerance Analysis

Pat Hammett

Interchangeability & Tolerances

n

Interchangeability --

or more components, manufacturers may

given the assembly of two

randomly select any sample for each component

and produce an acceptable assembly.

n

To insure interchangeability, manufacturers

assign tolerances to each component.

n

With interchangeability, a part dimension,

may deviate from its nominal specification,

d,

by some tolerance,

acceptable assembly.

t, and still produce an

N,

7

Types of Tolerances

n

Bi-lateral tolerance: Nominal +/-1mm

n

Most common in manufacturing

n

Unilateral tolerance: Nominal + 1 mm

n

Example: material thickness specification are often

one-sided because the goal is to use the least

amount of material (lower cost).

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Tolerance Analysis

Pat Hammett

Tolerance Analysis Approaches

n

Tolerance Allocation -(top-down)

n

assign tolerances of final assembly based on customer needs

n

work backward from assembly to components

n

Tolerance Synthesis -(bottom-up)

n

assign tolerances of components based on process capability.

n

work forward from components to assembly

n

Hybrid Systems (most common approach)

n

Assign initial tolerances from bottom-up --

tolerances are too large, -

tolerances are capable of meeting final assembly tolerances

work top-down until component

if final assembly

9

Tolerance Analysis Methods

Linear Stack-Up Analysis:

Case Stacking

tical Stacking -Root Mean Square Method

Non-Linear Stack-Up Analysis

tion

In this class, we will focus on linear stack-up

analysis.

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Tolerance Analysis

Pat Hammett

1. Worst Case Stacking

Join A & B

Part APart B

ASM AB = overall length

Worst Case Stacking --

extreme values for A & B.

assumes assembly AB must join

N

N

AB

ASM

=

∑

Nominal

i

+

∑

t

i

i

=

1

i=

1

Specification for Part A and B: 20 +/-0.5mm

Bottom-Up: if t

A

=t

B

= 0.5, then

t

AB

=t

A

+ t

B

= 0.5 + 0.5 = 1 >> t

AB

+/-1mm

11

Worst Case –Top Down Example

Join A & B

Part APart B

ASM AB = overall length

Worst Case Stacking --

extreme values for A & B.

assumes assembly AB must join

NN

AB

ASM

=

∑

Nominal

i

+

1

∑

t

i

i=i=1

Specification for AB: 40 +/-1 mm

Top-Down: if t

t

AB

= 1 then

AB

/ 2 = 1/2 >> t

A

=t

B

= +/-0.5 mm

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Tolerance Analysis

Pat Hammett

Statistical Tolerance Analysis

n

If two parts follow a probabilistic distribution,

then worst case stacking may be inappropriate.

n

For example, if the dimensions of part A and B

both follow a normal distribution, then the joint

probability of selecting the extremes is quite

small.

n

So, rather than using worst case, statistical

tolerance analysis is used.

13

2. Statistical Tolerance Analysis

Join A & B

Part APart B

ASM AB = overall length

If assume A & B follow probabilistic distributions such that

t

AB

=f (variation of A & B).

X=

∑

N

Nominal

N

=1

i

±

i

∑

t

2

i=1

i

Bottom-up

Tolerancing

t

22

AB

=t

A

+t

B

So ift

A

=t

B

= +/-0.7

t

2

AB

= .7

2

+.7 = 1

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Tolerance Analysis

Pat Hammett

Additive Theorem of Variance

Join A & B

Part APart B

ASM AB = overall length

n

Statistical tolerance: based on additive theorem of σ

2

.

Linear Stack-Up:

X

ASM

=

X

A

+

X

B

Mean Stack-Up:

X

ASM

= X

A

+ X

B

Variance Stack-Up:

σ

2

ASM

=σ

22

A

+σ

B

15

Example: Statistical Stacking

Join A & B

Part APart B

ASM AB = overall length

If Part A ~N(21, 0.20

2

) and Part B ~N(19, 0.15

2

)

What is the predicted assembly mean?

What is the predicted assembly sigma?

What are necessary tolerances for Part A and B to achieve 6σ?

(Hint: Suppose you need to achieve a Cp = 2.0.)

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Tolerance Analysis

Pat Hammett

Tolerances and Six Sigma

Join A & B

Part APart B

ASM AB = overall length

n

Suppose specification for ASM AB is 40 +/-1 mm

n

What sigma must be achieved for each component to

insure the assembly achieves a Cp = 2.0, Cpk> 1.5?

17

Worst Case Vs. Statistical

n

Suppose ASM tolerance +/1 mm

n

Worst Case Top Down: t

A

= t

B

= +/-0.5 mm

n

Statistical Top-Down: t

A

=t

B

= +/-0.7 mm

n

For the same assembly tolerance, statistical

statistical allows greater variation in the

components.

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Tolerance Analysis

Pat Hammett

3. Tolerance Simulation Models

n

The prior statistical stack-up analysis is based on root-

mean-squared method (additive variance theorem).

n

Alternatively, we could write a simulation model and

randomly generate combinations of components from

their distributions (e.g., normal, uniform, etc).

n

Then, compute the expected variance of the assembly.

Typically, you would then assign tolerances of say +/-

4σ

assy

or 6σ

assy

19

Bender Correction -Statistical

n

if more than 2 components are assembled

with statistical, we sometimes use correction

factors because components may become

unnecessarily wide.

n

Bender Factor, B = 1.5

t

assy

=B

∑

t

2

i

= 1.5

∑

t

2

i

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Tolerance Analysis

Pat Hammett

GilsonCorrection –Worst Case

n

If using worst case stacking, the Gilson

correction factor allows loosening of

component tolerances.

t

N

assy

=K

∑

t

K=

1.6

i=1

i

N

N2345

K1*0.920.800.72

if N=2; K (Gilson = 1.0)

21

Tolerance Analysis Examples

A

BC

D

overall length: ABCD = tolerance??

n

Bottom-Up Tolerance Stack-Up

n

If each block can be held to 20 +/-

expected tolerance for the assembly?

0.5 mm, what is the

n

Worst Case -Linear Stack

n

No Correction:

n

t

ABC

= +/-(.5+.5+.5+.5) = +/-2.0 mm

n

With Correction:

n

t

ABC

= +/-K(4*0.5) = .80(2.0) = +/-1.6 mm

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Tolerance Analysis

Pat Hammett

Non-Traditional Methods

n

Both worst case and statistical tolerance models

assume rigid parts and additive theorem of variance.

(often resulting in unnecessarily tight tolerances for

components within assembly)

n

Tolerance Adjustments:

n

Additional Factors: some models include assembly processing

variation in addition to the variation of joining components.

n

Weighted Tolerance Models: some models assign contribution

factors if one variable dominates the assembly.

23

Example: Non-Rigid to Rigid Part

n

A 2 mm thick center pillar is attached to an 0.7 mm thick

body side is 0.7mm thick with the following results:

Variable

Part A - Part B - Center Predicted Actual

Body SidePillarStack-UpAssembly

Mean-0.700.20-0.500.30

Std Deviation0.330.110.350.12

Why might you assign a tighter

Predicted Stack-Up

tolerance to the center pillar and a

X

ASM

= X

A

+ X

B

looser tolerance for body side?

σ

2

σ

22

ASM

=

A

+σ

B

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