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Dr Tafesse Gebresenbet
,
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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
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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.
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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.
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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
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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.
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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
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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.
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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.
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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)
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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,
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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 −+−+−+−+−=
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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
∑
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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
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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
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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
−=
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–
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.
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WC Tolerance analysis – Assembly gap
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 −=
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WC Tolerance analysis – Assembly gap
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 =
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i=1
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Worst Case Scenario Example
In this example, we see a mating hole and pin assembly. The nominal
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Worst Case Scenario Example
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Worst Case Scenario Example
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).
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Worst
Case
Scenario
Exam le
• Largest => 0.05 + 0.093 = 0.143
• Smallest => 0.05 ‐ 0.093 = ‐0.043
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Worst Case Scenario Example
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.
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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)
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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
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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.
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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 , .
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Root Sum‐of ‐Square
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
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Root Sum‐of ‐Square
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.
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Root Sum‐of ‐Square
, 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.
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Root Sum‐of ‐Square
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.
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Root Sum‐of ‐Square
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 ‐ .
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R S f S
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Root Sum‐of ‐Square
.
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 ++++=
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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
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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.
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Pool Variance in RSS
,
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.
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P l V i i RSS
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Pool Variance in 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.
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l i i
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Pool Variance in RSS
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
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D i RSS
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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
Dr. Tafesse Gebresenbet
p pi i=1
42
N li RSS
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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
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RSS Example
• Largest => 0.05 + 0.051 = 0.101
• Smallest => 0.0 ‐ 0.0 1 = ‐0.001
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RSS Example
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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.
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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
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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
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Quality Loss Function
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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
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Quality Loss Function
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.
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Cost of Off Target and Sensitivity
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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
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Cost of Off Target and Sensitivity
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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 .
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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
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