tolerance design and analysis part iii

8/9/2019 Tolerance Design and Analysis Part III

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Dr Tafesse Gebresenbet

,



References

1. reve ng, . ., o erance es gn; a an oo or eve op ng optimal specifications,1996, Addison Wesley

2. Fortini Dimensionin for interchan eable Manufacture

1967,Industrial press3. Ken Chase, Basic tools for tolerance analysis of Mechanical

ssem es, 2001

4. Bjorke,

Computer

aided

tolerancing,

2nd

edition,

1992,

Library

of on ress

5. Kai Yang & Basem Al Haik, Design for Six Sigma: A road map for product development.2003, McGraw‐Hill

6. Paul J. Drake, Jr., Dimensioning and Tolerancing Handbook, McGraw‐Hill, 1999

Dr. Tafesse Gebresenbet 2



Tolerance Design

o erance es gn s a out t e eng neer ng

process

or

developing tolerances.

tolerances

to

the

part,

assembly,

or

process,

identified

in the functional and physical structures, based on

overa to era e variation in FRs, t e re ative in uence o

different sources of variation on the whole, and the cost bene it

trade

‐o s.

This step calls for thoughtful selection of design

parameter

tolerances and material

upgrades that will

be later cascaded to the process variables.

A tolerance must be developed before it can be commun ca e , s s w a we ca o erance es gn.




Tolerance Design

n eng neer ng eam mus o ow a me o ca oug u

approach develop the tolerances for component parts, subassemblies, and a total system by considering the internal

,

reliability

growth. Leaving this thoughtful and methodical exercise to little more

an an e uca e guesses us pr or o pro uc on eaves e engineering process open to every individual personal opinion

and it is prone to error. W en to erances are not we un erstoo , t e ten ency is to over

specify with tight dimensional tolerances to ensure functionality and thereby incur cost penalties.

Traditionally, specification processes are not always respected as credible. Hence, manufacturing and production individuals are tem ted to make u their own rules.




Tolerance Design

Joint efforts between design and process in the team help improve understanding of the physical aspects of tolerance and

‐

balanced

e process y w c a pro uct s to erance requ res t e

considerations of;

Selection

is

based

on

the

economics

of

customer

satisfaction, ,

the cost of manufacturing and production, and

the relative contribution of sources of FR.

When this is done, the cost of the design is balanced with the

quality of the design within the context of satisfying customer




Tolerance Design

y

eterm n ng

w c

to erances

ave

t e

greatest

impact on FR variation, only a few tolerances need to

be ti htened, and o ten, man can be relaxed at a

savings.

The

quality

loss

function is the basis for these decisions.

The proposed process also identifies key characteristics , variability reduction will result in corresponding customer benefits.




Tolerance Design

or a pro uc or serv ce,

cus omers

o en

ave

exp c

or

mp c

requirements and allowable requirement variation ranges, called

customer tolerance.

,

into

design

functional

requirements

and

functional

tolerances.

For a design to deliver its functional requirements to satisfy functional ,

and their variations must be within design parameter tolerances.

For design development, the last stage is to develop

manufacturing

process .

Tolerance development stage




Tolerance Design

os o e wor n o erance

es gn

nvo ves

e erm na on

o

es gn

parameter tolerances and process variable tolerances, given that the

functional tolerances have already been determined.

e erm o erance es gn n mos n us r es ac ua y means e

determination of design parameter tolerances and process variable

tolerances.

If a product is complicated, with extremely coupled physical and

process

structures,

tolerance

design

is

a

multistage

process.

After

functional

tolerances

are

determined,

system

and

subsystem

tolerances are determined first, and then the component tolerances are

determined on the basis of system and subsystem tolerances.

Finall , rocess variable tolerances are determined with res ect to

component

tolerances.

A typical stage of tolerance design.




Tolerance Design

n a typ ca stage o to erance es gn, t e ma n tas s, g ven t e target requirement of y and its tolerance (i.e, T y±Δ0 ) , how to assign the tolerances for xi values. There are three major issues in tolerance

1. Manage variability.

2. Achieve functional re uirements satisfactoril .

3. Keep life‐cycle cost of design at low level

In tolerance design, cost is an important factor. If a design parameter or a process variable is relatively easy and cheap to control, a tighter tolerance is desirable; otherwise, a looser tolerance is desirable.

Therefore, for each sta e of tolerance desi n, the ob ective is to

effectively ensure low functional variation by economically setting appropriate tolerances on design parameters and process variables.




Tolerance Design

ere are two c asses o to erance es gn met o s:

Traditional tolerance design methods include

‐ ,

statistical tolerance analysis, and

cost‐based tolerance analysis (not included in this lecture series)

the relationship between customer tolerance and producer’s tolerance, and

tolerance design experiments. (not included in this lecture series)




Worst Case

Tolerance

Anal sis

Extreme or most liberal condition of tolerance buildup

“…tolerances must be assigned to the component parts of t e mec anism in suc a manner t at t e pro a i ity t at a mechanism will not function is zero…”

Worst‐case‐anal sis is not considered a statistical procedure but is used often for tolerance analysis and

allocation.

s met o prov es a as s to esta s t e mens ons and tolerances such that any combination will produce a functionin assembly.

This method compares the part tolerances with the entire assembly tolerances to reveals the extreme or most liberal

“con on o o erance u ‐up; ence, e erm wors ‐

case”.

Evans

describes

it

as,




Worst Case

Tolerance

Anal sis

The general model for the WC analysis is the sum of all component dimensions at their WC maximum and

m n mum va ues

ntotal d or or d or d or d d )).....(()()( 321 −+−+−+−+=

The assembly will likely have to fit within a space defined

by

another

feature

on

another

component

or

assembly.

, mat ng.

To define the gap between the mating dimension, the

o ow ng a ustment must e ma e to t e genera

tolerance model

]))......(()()([ 321 nmatinggap d or or d or d or d d d −+−+−+−+−=




Worst Case Tolerance Analysis

maximum size is expressed

m

Where WC max = N p i

+ T p i

( )i=1∑ Npi – the nominal or target value of the component dimension of

any part

Tpi – the initial tolerance assigned to Npi m‐total number of parts in the assembly

The WC minimum dimension can occur for the entire assembly

WC min = N p i− T p i

(m

∑

Dr. Tafesse Gebresenbet

i=1

14



Worst case tolerance analysis

ntro uc ng new terms re err ng to t e geometry as t e enve op an

remaining consistent with the 6σ institute terminology

Ne – normal envelop,

Te‐ the envelop of tolerance

Q‐ the minimum gap constant or smallest allowable gap

∑=

+−−≤m

i

pi piee T N T N Q

1

)(

In some applications we will use Q= 0, which means we are technically at the point of a line‐to‐line fit. As Q gets bigger we

will see a finite growth in the assembly gap

N e ‐ T e

=> Minimum assembly envelope




Worst case tolerance analysis

The upper boundary on the allowable assembly gap is quantified as:

RT N T N

m

i

pi piee ≤−−+ ∑=1

)(

In many cases the assembly gap could get too big for the proper functionality of the assembly within the envelope. R

is the u er bound on the assembl a . and R are technically related to the assembly process and are thus critical boundary for it

Ne + Te => Maximum assembly envelope




WC Tolerance analysis – Assembly gap

Let us re resent G‐ Assembly gap

Gmax – the maximum assembly gap, which is equal to R

Gmax = N e + T e − N p i − T p i( )i=1

m

∑ The minimum assembly gap is defined by Gmin

− −m

min e e p i p i

i=1

max

min

acceptable range of assembly gap, Grange

−=


m nmaxrange

18



–

The maximum worst‐case condition, WC max, and

t e m n mum worst‐case con t on, cm n, are expressed here mathematically.

ow a we now e ex remes an nom na gaps, we can begin to allocate the tolerances so that the

calculated tolerance conditions.





can now define the nominal assembly gap that is the desired

target for the assembly

( )∑=−=

m

i

pealno i N N G1

min

We can do some rearranging to make it easy to calculate the nominal envelope dimensions

m

The final gap dimension we are left to define is the tolerance on

∑=

+++=i

e pi pie T QT N N 1

)(

the gap, Gtolerance

minmax GG −=


2o erance

20




s ng otora a s un que convent on to account or ot

magnitude and directionality of a dimension (for complex assembly)

Where the term Vi is used to define a unit vector that quantifies the algebraic sign of the component as it is stacked in the assembl , we et

∑=

=i

iialno V N G1

min

ccount ng or , w ere t e true pos t on w t respect to a datum is considered; the tolerance zone must be converted from

diameter to a radius when used for the vectorized summation

process. A correction factor Bi is added in the equation to account for it [Harry & Stewart,1988]

m

BV N G =


i=1

21



Worst Case Scenario Example

In this example, we see a mating hole and pin assembly. The nominal


.

22




Here, we can see the two worst case situations where th ins ar in th xtr m out r d s or inn r d s.

The tolerance stack up can be evaluated as seen here.

In this exam le the anal sis be ins on the ri ht ed e of the right pin.

You should always try to pick a logical starting point for stack analysis.

Note that the stack up dimensions are summed

accor ng

o

e r

s gn

e

arrows

are

e

displacement vectors).




Worst

Case

Scenario

Exam le

• Largest => 0.05 + 0.093 = 0.143

• Smallest => 0.05 ‐ 0.093 = ‐0.043





From the stack up, we can determine the tolerance calculations as seen in this table.

Analyzing the results, we find that there is a +0.05 nominal gap and +0.093 tolerance buildup for the worst case in t e positive irection.

This gives us a total worst‐case largest gap of +0.143. It ‐ .

interference fit.

Thus in this worst‐cas sc nario th arts will not fit

and one needs to reconsider the dimension or the tolerance.




Nonlinear Tolerance stacks using the WC method or cases w ere t e re at ons p n t e component parts n an

assembly are not going to stack up in a linear relationship with respect to the assembly gap or other geometric‐dependent variables.

us e r s n epen en assem y var a e an e n epen en

component variables: y = f ( x1, x2, x3,... xn )

In considering this facts in WC Tolerance analysis with non‐linear

problems (Greenwood & chase) formulated by considering small changes in the assembly or dependent dimension may be expressed by a Taylor’s expansion series as follows:

2

The common ractice for tolerance anal sis is to substitute tolerances

...2 1 1+ΔΔ∂∂+Δ∂=Δ ∑∑∑ = =

ji

i j ji

i

i

y x x x x x x

for the delta quantities (See for similar formulation Fortni, 1967)




Nonlinear Tolerance stacks using the WC method

In some cases, the relationship between the components and the assembly will stack up in a linear

.

When it occurs, one must determine a function that can be used to define the relationshi between the dependant assembly variable, y, and the independent component variables, x sub n.

the nominal dimension and tolerance can be used as shown here.

The partial derivatives represent the particular sensitivity that each component dimension and




Nonlinear Tolerance stacks using the WC method

Greenwood and Chase defined a general case WC tolerance equation as follows:

Tol y =

∂ f

∂ x1tol1 +

∂ f

∂ x2tol2 +

∂ f

∂ x3tol3 + ...+

∂ f

∂ xntoln

Nom y ≈ ∂ f

∂ x1

x1 + ∂ f

∂ x2

x2 + ∂ f

∂ x3

x3 + ...+ ∂ f

∂ xn

xn

In a nonlinear problem we must define specific values for

t e

part a

er vat ves

s

t ey

represent

t e

part cu ar

sensitivities that each component dimension and tolerance will induce on the assembly dimension and tolerance.




Root Sum‐of ‐Square

Although useful for interchangeability, the worst case analysis, however, is also very conservative and does no ensure 100 pro uc y.

The worst case scenario does not take into account the .

For example, suppose the probability for a defect is 10

% and there are arts in a linear s stem.

The probability or chances of producing a system with

all five “out‐of ‐spec” parts is 0.10 raised to the 5th

which equals 0.001 percent. The such a remote chances of defect, one could

v y , .





The real world of component manufacturing is dominated

by the laws of probability, random and chance and special

on

target

every

time)

Statistical tolerance analysis assumes that all processes are in control, so all estimates of tolerance accumulation are

1 − 1/ 2 x− /σ 2

x = σ 2π e





Random variations caused by external or deteriorative sources that moves the process from a state of random

of

variability

that

is

beyond

what

is

occurring

due

to

natural (random) events are represented by batch‐to‐batch

ar a y, amage or worn oo s, con am na e raw

materials, and numerous other noises.

In Ta uchi aradi m, noises are alwa s active in

manufacturing processes and SPC charts are viewed as measuring external, unit‐to‐unit, and deterioration noise

.

If one noise source is intensified or is somehow initiated

within the manufacturing sources of variability that can be corrected during the manufacturing process.





, be considered to account for the more likely chances of having dimensions which do not all occur at the extreme limits simultaneously.

The root mean square utilizes basic statistical methods to a t e measure o varia i ity.

We

postulate

a

probability

distribution,

as

seen

here,

for

eac component n t e assem y s a ran om var a e, f(x) is the probability function of x and mu and sigma are constant parameters.





occur between the mating assemblies.

‐

transform or student t‐transform to calculate probability of assembly success.

The sum of squares is a mathematical treatment of the data

to

facilitate

the

legitimate

addition

of

measures

of

ar a ty

RSS method is used to add up tolerance stacks when more .

assemblies must then fit into another assembly.





The sum of squares is a mathematical treatment of the data to facilitate the legitimate addition of measures of variability

two

components

are

assembled

together.

Often

these

assemblies

must then fit into another assembly.

RSS is used to determine if a functional fit is going to occur b/n a probability of assembly success or failure can be calculated using

Z‐transform for population data, or by using the student t‐transform for sample data.

This allows us to study the joint probabilities relative to the ‐ .


R S f S




.

component has a measured average and standard deviation

associated with it due to variation in the manufacturing process.

The sum or differences of each of these average component dimensions will define the avera e dimension

nd or or d or d or d d ~

))....((~

)(~

)(~~

321 −+−+−+−+=

The sum of the squares of each of the dimensional standards

deviations

defines

the

overall

assembly

22

3

2

2

2

1

2 .... nsssss ++++=




RSS method

In statistics, it is arithmetically wrong to simply add

the standard deviations linearly.

,

squares

of

the

variances

and

taking

the

square

root.

Similarly, tolerance stacking works in a similar fashion.

Assembly tolerance stack equation

f ( x) = T 12 + T 2

2 + T 32 + ...T n

2

Tn – represents the nth component tolerance in the assembly




Pool Variance in RSS

The adjusted standard deviation if the actual process standard deviation is not known is given is determined:

σ = Tol

p

the standard deviation.

For normal distribution with an unknown standard

deviation, one would use the adjusted standard deviation

ca cu a ons w c cons ers e capa y va ue, p.





,

assembly is assumed to be independent of every other component variance; thus, each component will possess its own p va ue.

This allows us to formulate a more realistic calculation of the variances called oolin of the variances which results in the sigma gap formula seen here.

T ⎛ ⎞2

T ⎛ ⎞m 2

σ gap =3Cp⎝ ⎠

+3Cpi⎝ ⎠i=1

rom e ormu a, one can see a e s an ar ev a on

of the gap assembly is expressed as the square root of the pooled variances from the envelope the assembly is to fit

within and the sum of the component variances.


P l V i i RSS




‐

the following equation, where the the Z‐transform is used

to calculate the actual probability of exceeding the limits of the gap.

Z Q = Q − G

nom

σ gap

which is an interference fit.

In the formula seen here, ZQ is determined when Q is

su tracte

rom

t e

nomina

gap

an

compare

to

t e

standard deviation of the assembly gap.


l i i




the mating assembly.

In this case, we assume that Q=0, in order to have an

interference fit represented by line‐to‐line contact.

Thus, ZQ is simply the number of standard deviations away

interference. This Z value is then used to look up the

corresponding

probability

in

the

Z‐

chart.

Q − N e − N pi

m

∑

⎜ ⎟

Z Q =

=

T e

3C

⎛⎜

⎞⎟

2

+ T pi

3C

⎛⎜

⎞⎟

m

∑2


=

41

D i RSS



Dynamic RSS

n a s m ar as on, one can ca cu ate t e va ue or the maximum condition in which we exceed the

and

Gmax to

determine

the

Z

values

at

the

gap

limits.

Z G min = min − nom

T e

⎛ ⎞2

+ T

pi

⎛ ⎞2m

3Cp 3Cpi i=1

Z G max = max − nom

T e⎛⎜

⎞⎟

2

+ T pi⎛⎜

⎞⎟

2m


p pi i=1

42

N li RSS



Nonlinear RSS

on near oot sum o squares eterm nat on s similar to the linear methods only the tolerance is

for

the

adjusted

sigma.

Tol y =

∂ f ⎛

⎜

⎞

⎟

2

tol1

2

+

∂ f ⎛

⎜

⎞

⎟

2

tol 2

2

+

∂ f ⎛

⎜

⎞

⎟

2

tol3

2

+ ...+

∂ f ⎛

⎜

⎞

⎟

2

toln1 2 3 n

σ adjusted = o i

3Cpk i




RSS Example

• Largest => 0.05 + 0.051 = 0.101

• Smallest => 0.0 ‐ 0.0 1 = ‐0.001


RSS Example



RSS Example

before and consider the RSS method of analysis.

nominal

gap

and

a

+0.051

tolerance

buildup

for

the

RSS case in the positive direction.

For the largest gap, we would have a total gap of +0.101. When at the smallest gap, the result is a ‐0.001which is

.

Although, technically, this scenario would not work

scenario case.




Ta uchi Method

Input from the voice of the customer and QFD processes

Select proper quality ‐loss function for the design

Determine customer tolerance values for termsin Quality Loss Function

Determine cost to business to adjust

a cu ate anu actur ng o erance

Proceed to tolerance design




Taguchi

The first step in the process include getting input from the parameter design. The voice of the customer and Quality

insight into the customer expectations. Other inputs from

the parameter design include:

p mum con ro ‐ ac or se po n , o erance es ma es determined from engineering analysis, and initial material

grades

selected.

Voice of customer

Quality function deployment

Optimum control‐factor set points Tolerance estimates

Initia materia gra es


Quality Loss Function




The first step in the process include getting input from

the parameter design. The voice of the customer and

Quality function Deployment processes help provide eta e ns g t nto t e customer expectat ons. t er

inputs

from

the

parameter

design

include: Optimum control‐factor set point, tolerance estimates

e erm ne rom eng neer ng ana ys s, an n a

material grades selected.

Identify

customer

costs

for

intolerable

performance

L( y) = k ( y − m)2 = Ao

Δ o

( y − m)2





The quadratic loss function is described here. In the formula, L(y) is the loss in dollars due to a deviation away from the target

, , product; m is the target value of the product’s response; and k is an economic loss function called the the quality loss coefficient and is calculated as A zero over Delta zero s uared. .

The typical quality loss function is also illustrated here. From

the figure, one can see that at y=m, the loss is zero; the loss

increases as moves from m. As the curve approaches the customer tolerance limits, Delta

zero, the cost for the poor performance increase.

, .

function is shown here for example purposes and is not indicative of all quality loss function behavior; the quality loss function can and would chan e de endin on the customer’s tolerance and usage environment.


Cost of Off Target and Sensitivity




The next step, one needs to determine the cost to the business to adjust the off ‐target performance values

,

and the sensitivity between the customer tolerance and manufacturing tolerances.

Cost to business to adjust off target performance

Sensitivity, β

φ = o A A = o

Δ [ β ( x − m)]2





Cost of Off Target and Sensitivity Once the function and its limits are established the

engineering team need to determine the a safety factor, phi, to prevent off ‐target performance values.

The company also needs to quantify what expense is worth

funding to remedy the off ‐target performance.

Therefore, the safety factor can be described as the square root of the average loss in dollars when a product c aracteristic excee s customer to erance imits over t e average loss in dollars when a product characteristics exceeds the manufacturing and/or design tolerance limits.

ens t v ty s t e c ange n t e g ‐ eve customer observable characteristic or a product‐level engineering characteristic, y, when a unit change occurs from the target

. The relationship of the of the sensitivity in the

manufacturing loss, A, is equal to the loss function for .




S



Summary

Accounting for the safety factor and the sensitivity, we can

link the customer limits to the development of design

element limits. The result is delta, the manufacturing

tolerance based on the Taguchi equation.

, , forward to the design process where further tolerance design can be optimized.

Note that we discussed three methods to establish

tolerance

‐ ,

statistical tolerance analysis, and

Taguchi tolerance design methods


tolerance design and analysis part iii

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