Tolerance Design - University of Rochester · Tolerance Design PD Funkenbusch (ME 222/424) 2 Where...

ME222/424

MSC 424

TME 424

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Tolerance Design

PD Funkenbusch (ME 222/424, TME 424, MSC 424)

Tolerance Design

PD Funkenbusch (ME 222/424)

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Where do tolerances come from? History

Absolute vs. Statistical tolerance as acceptable range

Summing tolerances Worst-case

Statistical

More complex systems

Tolerance Design methodology Tolerancing based on variance control

Interchangeable parts 3

Inventor? Depends on your interpretation. But maybe as early as 1040’s AD (Chinese moveable type).

Eli Whitney helped popularize the idea for the manufacture of guns.

“Took off” in the 1800’s as a cornerstone of mass production.

Tolerances make sure parts will fir together.

http://3.bp.blogspot.com


http://p2.img.cctvpic.com

Tolerance as acceptable range


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Set maximum and minimum values for some characteristic (e.g. part length). LSL = Lower Specification Limit USL = Upper Specification Limit

For simplicity we will assume that the average value (m) is midway between the LSL and USL.

Tolerance = D LSL = m - D

USL = m + D

Range of values m ± D

m ± D


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User (customer) friendly

Immediate sense of the values likely to be encountered.

Quick measure of “quality”

Small D values are immediately impressive

But what does the tolerance actually mean?

Two common methods to specify

Absolute

Statistical

Absolute (method one)


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Inspect all components

Reject (discard) all those outside of the tolerance (m ± D)

All parts (that pass) will be within the tolerance

Absolute limit

Statistical (method two)


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Analyze a sample of components. Determine average (m) , standard deviation (s), and distribution.

Set tolerance so that only a small, specified fraction of components

will be outside of the range.

For simplicity here, we will assume that all distributions are normal. Examples:

D = 1 s 68.3 % in tolerance average ± one standard deviation D = 3 s 99.73 % in tolerance D = 6 s 99.9999998% in tolerance

D = 4.5s 99.99966% in tolerance

Note: D = 4.5s (evaluated over an extended time period) is sometimes used as the

cut-off point for “Six Sigma Quality” less than 3.4 DPMO (defects per million opportunities) “rule of thumb” to adjust for drift in product mean

Summing tolerances


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Often concerned about how to sum tolerances.

Determine final tolerance based on component tolerances

Determine how to adjust component tolerances to achieve a desired final tolerance Add in information on costs

to find the most cost effective approach

m1 ± D1

m2 ± D2

? ± ?

m4± D4

Example


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Fit three components end to end into a slot machined into a fourth component.

Want to determine the “gap” remaining.

Average length of gap

Tolerance on the gap

m2 ± D2

m2 ± D2

m3 ± D3

m1 ± D1

mg ± Dg

Gap


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Average add & subtract based on

geometry

mg = m4 - m1 - m2 - m3

Tolerance always sum Uncertainty increases with each

component included

Different ways of summing “Worst case” “Statistical”

mg ± Dg

“Worst-case” summation


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With absolute tolerances, this ensures that all the gaps will be within the tolerance (i.e. mg Dg )

Can also use with statistical tolerances. In this case it is possible to have product outside of tolerance but the probability will be small.

mg ± Dg

Tolerance sum D’s

Dg = D1+ D2 + D3 + D4

“Statistical” summation


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Most meaningful with statistical tolerances.

Assuming all distributions are normal and that the component tolerances are set to the same number of s’s (e.g. D = 3s for all components), then the product tolerance should correspond to that of the components.

mg ± Dg

Tolerance add D2 ‘s

Dg2 = D1

2 + D2

2 + D32 + D4

2

Where does this (summing squares) come from?


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From statistics, know that uncorrelated variances can be added:

stotal2 = s1

2 + s2

2 + s32 + s4

2 …

Multiply thru by a constant squared, n2:

n2stotal2 =n2s1

2 +n2s2

2 +n2s32 +n2s4

2 …

But D = ns is how we defined the statistical tolerance, so:

Dproduct2 = D1

2 + D2

2 + D32 + D4

2 …

m4± D4

Numerical example


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Fit three components end to end into a slot machined into a fourth component.

m2 ± D2

m2 ± D2

m3 ± D3

m1 ± D1

mg ± Dg

Comp. m (mm) D (mm)

1 10 0.1

2 30 0.3

3 20 0.3

4 61 0.2

Average gap


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Add & subtract based on geometry

mg = m4 - m3 - m2 - m1

mg

= 61 – 20 – 30 – 10

= 1 mm

mg ± Dg

Comp. m (mm) D (mm)

1 10 0.1

2 30 0.3

3 20 0.3

4 61 0.2

Worst case Statistical


Dg = D1+ D2 + D3 + D4

=0.1 + 0.3 + 0.3 + 0.2 = 0.9 mm

mg Dg 0.1 to 1.9 mm

Dg2 = D1

2 + D2

2 + D32 + D4

2

= (0.1)2 + (0.3)2 + (0.3)2 + (0.2)2

= 0.23

Dg = 0.5 mm

mg Dg 0.5 to 1.5 mm

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Tolerance on gap

Tolerance type vs. summation method


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Summation method

Tolerance type

Absolute Statistical

Worst-case (add D’s)

All product within tolerance

Some product out of tolerance (generally

small %)

Statistical (add D2 ’s)

Difficult to estimate (“depends”)

Fraction in tolerance related to components’

fraction in tolerance

Complications


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Simple approach works well for a linear stacking of components as shown in the example.

May not apply to: components with other “features” shape imperfections, roughness… more complicated geometries 3-D, rotations… properties other than length variability caused by other sources (e.g. environmental conditions) etc.

As examples, variability in the output voltage of an electrical circuit because of differences in

component properties (resistances, capacitances, etc.) and geometry/assembly variability in engine performance due to component wear, ambient temperature,

fuel quality, etc.

Approaches to dealing with more complex systems


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Modeling Need a good mathematical model May be able to solve analytically, depending on complexity Alternatively use the model to “test” different combinations of

component /environment values Monte Carlo random sampling based on frequency of occurrence.

Generally requires large numbers of samples (1,000s or 10,000s) must be practical with the model

Various systematic approaches, e.g. Tolerance design

Experimental Need to be able to identify and monitor or adjust sources of

variability Analyze data collected “in the field” is suitable data available? Systematic testing, e.g. Tolerance design

Random sampling (Monte Carlo) example


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Simple model for axial loading of a long bone. [This is taken from a “Case Study” in Bartel, Davy, and Keaveny’s Orthopedic Biomechanics textbook. However, it is just an isostrain model for uniaxial compression.]

cortical

trabecular t

D

P

Random sampling (Monte Carlo) example


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The load supported on the cortical portion of the bone,

Pc, is given by:

Where P is the total applied load, t is the thickness of the cortical shell, D is the diameter of the trabecular centrum, and Ec and Et are the modulus of the cortical and trabecular bone, respectively.

Because of each of the terms in this equation will vary from person to person, the load on the cortical bone will also vary.


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The table below gives some information on each of the parameters.

Average values (except for P) are taken from Bartel et al. and are nominally for vertebra. The rest of the values are “stand-ins” (more or less made up), just to illustrate.

Parameter Average value Coefficient of variance

P (N) 1.5e3 20%

Et (Pa) 3.0e8 20%

Ec (Pa) 1.7e10 10%

t(m) 3.5e4 25%

D (m) 3.0e-2 25%

Average Mean Square Error


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Results (different each time!)

9.50E+02

1.00E+03

1.05E+03

1.10E+03

1.15E+03

1.20E+03

1 10 100 1000 10000

Lo

ad

(N

)

N

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

1.20E+05

1.40E+05

1 10 100 1000 10000

MS

E (

N*N

)

N

Average 1.09 e3 MSE 7.09 e4


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Compare with a tolerance design (8 TC)

9.50E+02

1.00E+03

1.05E+03

1.10E+03

1.15E+03

1.20E+03

1 10 100 1000 10000

Lo

ad

(N

)

N

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

1.20E+05

1.40E+05

1 10 100 1000 10000

MS

E (

N*N

)

N

T O L E R A N C E A S V A R I A B I L I T Y


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Tolerance Design

Taguchi’s approach to quality


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Genichi Taguchi (1924 – 2012) quality “guru”, responsible for many innovations developed many tools and methods Tolerance design was Taguchi’s last resort method for improving

quality

Taguchi’s concept of quality Taguchi equated “quality” with reducing the variance (s2) in the final

product Didn’t believe in using fixed “tolerances” (i.e. cutoff values) So Tolerance design focuses on reducing s2 , without considering %

in/out of tolerance Can be applied to non-normal distributions, but need to be cautious

about converting to a “D” and estimating % in tolerance

Tolerance Design concept


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Assume that proportionality between variance in components and final (product) variance still holds, but with a proportionality constant (sensitivity) added

stotal2 = h1s1

2 + h2s2

2 + h3

s32

+ h4 s4

2 …

Experiment estimate variance for the product

determine contribution of each component variance to the total decide how to best improve tolerance (i.e. reduce variance) as needed

h values show sensitivity of final product variance to tolerance (variance) of each component think about the units…

Tolerance design experiment


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TC

A

B

C

...

measured

response

1 -1 -1 -1 Y1

2 -1 -1 +1 Y2

3 -1 +1 -1 Y3

4 -1 +1 +1 Y4

... +1 -1 -1 …

For each component, input specific values match variance of component (“levels” -1 ,+1) m ± s

Experiment tests different combinations of component levels

Measure the response of the product variation in these values

provides estimate of the total product variance.

also determine contribution of each component to total

TC = Treatment condition, one “run” of the experiment A, B, C = different components -1, +1 = represent two different component values to be used in experimentation

Matrix selection


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Design of matrix is important Design Of Experiments (DOE)

Usually 2-level

Can include other (non-component) sources

Matrix size (# of TC) Between ~ (n+1) and 2n

n = number of components

Much smaller than Monte Carlo style methods

Large matrix provides more/better data (rare)

But smaller sizes are still useful (common)

Example (Throttle handle)


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From “Designing experiments for tolerancing assembled products”, Soren Bisgaard. Technometrics (1997), 142-152

Friction in a throttle handle of outboard motors too much or too little need to improve the tolerance

Tracked friction by measuring torque to turn the handle

Three components in the assembly

Components Knob, handle, and tube But multiple dimensions on

the knob (three), and handle (three)

Total of seven dimensions to tolerance

Matrix size Minimum size 8 Maximum size 128 Chose to use 64

Relatively conservative/expensive

Throttle handle (experimental details)


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Knobs (dimensions A, B, C) m - s and m + s for each dimension Eight possible combinations Manufactured all eight combinations

Handles (dimensions D, E, F) m - s and m + s for each dimension Eight possible combinations Manufactured four of the combinations

Tube (dimension G) Manufactured two tubes, one with m - s and one with m + s

Tested all combinations of these components 8 x 4 x 2 = 64 combinations 64 TC Treatment condition assembled one combination of components and

measured torque

Throttle handle (key results)


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A, B, C knob dimensions

D, E, F handle dimensions

G tube dimension

G is main contributor to variance (>50%) Best bet to improve

performance

But also depends on relative costs

Source s Variance (contrib. to stotal

2)

%

A 0.0028 29.50 4.36

B 0.0023 56.91 8.40

C 0.0024 10.04 1.48

D 0.0037 107.33 15.83

E 0.0030 45.16 6.67

F 0.0043 86.27 12.74

G 0.0040 342.25 50.52

Total -- 677.46 100.00

Part tolerances (length)

Variance of torque (force-length)2

Throttle handle (predicting improvement)


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Consider the effect of halving the tolerance (i.e. s) for G Variance of G (s2 ) will be

reduced to ¼

Contribution from G to total with, therefore also be reduced to ¼

Total Variance for the throttle torque should be reduced to ~ 421

677 – ¾ (342) = 421

Source s Variance (contrib.

to D2)

%

A 0.0028 29.50 7.01 B 0.0023 56.91 13.52 C 0.0024 10.04 2.38 D 0.0037 107.33 25.49 E 0.0030 45.16 10.73 F 0.0043 86.27 20.49 G 0.0040

0.0020 342.25

85.56 20.32 Total -- 677.46

420.77 99.95

Predictive equation


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stotal2 = hA

sA2

+ hBsB2

+ hCsC2

+ hD

sD2

…

hG sG

2

= contribution of G to total = SSG = 342.25

hG

= 342.25/ sG2

= 342.25/(0.0040) 2

= 2.14 x 10 7

Reduce sG to 0.0020

SSG= hG sG

2

=2 .14 x 10 7 x (0.0020) 2 = 85.56 Part tolerances

(length) Variance of torque (force-length)2

Source s Variance (contrib.

to D2)

%

A 0.0028 29.50 7.01 B 0.0023 56.91 13.52 C 0.0024 10.04 2.38 D 0.0037 107.33 25.49 E 0.0030 45.16 10.73 F 0.0043 86.27 20.49 G 0.0040

0.0020 342.25

85.56 20.32 Total -- 677.46

420.77 99.95

Summary

Definitions of tolerance Based on % in/out of tolerance Absolute all in tolerance Statistical known % out of tolerance

Summation of tolerances “worst-case” summation of tolerances “statistical” summation of the squares

Tolerance design Based on reducing the product variance Assumes product variance is proportional to component variances DOE to estimate total product variance, component contributions,

and the effects of changing component tolerances

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Tolerance Design - University of Rochester · Tolerance Design PD Funkenbusch (ME 222/424) 2 Where...

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