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Session #1 : The Basics

� Objectives:

� How to calculate mean dimensions with equal

Bilateral Tolerances

� Calculating Inner and Outer Boundaries

� Virtual and Resultant Conditions

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What is Tolerance Stack-up Analysis?

Tolerance Stack-up Analysis (also called as Gap

Analysis, Loop Diagrams or Circuit Analysis) is the

process of calculating minimum and maximum

airspaces or wall thickness or material interferences in

a single part or assemblies

It’s a logical process broken in few steps…

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Steps in Tolerance Stack-up Analysis

� Step #1:– Identify objectives: for example, you want to test if no

interference is possible at a certain place in an assembly, then

you set your requirement as “Gap must be equal to or greater

than zero”

� Step #2:– Identify all dimensions that contribute to your objectives as

defined in step #1 (gap) and convert them to equal bilateral

toleranced dimensions; if they are not already

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� Step #3:– Assign each dimension a +ve or –ve value. For Radial stacks

(going up and down); start at the bottom of gap and end up at the top of gap

– Down direction is –ve (top of gap to bottom)

– Up direction is +ve (bottom of gap to top OR towards end)

– Stacks that go left and right in the assembly, start at the leftside of gap and end up at the right side of the gap.

– Left direction is –ve (right of gap to left)

– Right direction is +ve (left of gap to right OR towards end)

� Remember that you are working one part at a time; so deal with o nepart’s significant features before jumping to next part. This is the best way to work with assemblies having many parts

Steps in Tolerance Stack-up Analysis …

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� Step #4 (Basic Rules):– Remember that one set of mating features between parts creates the variable

you are looking for. Variable in this case is either minimum gap or maximum gap or maximum overall assembly dimension. One set mating featur es creates it. So, though multiple routes may have to be investigated to fi nd this most significant set of features, only one set creates worst case, fr om one part to next.

– Its often mistake to follow one route from one set of mating fea tures(holes/shaft, hole/pin) then continue the same route through ano ther set. One of these sets creates the smallest or biggest gap or maximum ove ralldimension, Once you find, which it is, others become non-factors in analysis.

– Using more than one set of features within same two parts, will most likely produce wrong results. Still tolerances from other features may contribute to the critical set you are using. For example: when datum features arereferenced at MMC or when more than one set of datum features co me into effect.

Steps in Tolerance Stack-up Analysis …

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� Step #5 (Basic Rules):

– When a single feature or a pattern of features are controlled by

more than one Geometric Tolerance (such as orientation

combined with position), the designer must determine which, if

either is contributing factor to variable. It is also possible that

none of geometric tolerance is a factor and instead size

dimensions are factors.

– The Designer must deduce what factors are pertinent through

sketches and reasoning.

– The judgment of designer is critical in these determinations.

Steps in Tolerance Stack-up Analysis …

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� Add all +ve and –ve dimensions which will calculate your mean gap. If mean gap is –ve number, your requirement of ‘no material interference’ (or ‘clearance’ in other words) is already violated!

� Then we must add sum of equal bilateral tolerances (1/2 of total tolerance) to the mean dimension (or gap) to determine maximum gap.

� Then we must subtract the sum of equal bilateral tolerances (1/2 of total tolerance) from the mean dimension (or gap) to determine minimum gap.

� Again any –ve value for minimum or maximum gaps indicate interference situation

� Maximum gaps are maximum clearance (or in case of interference fits, minimum interference)

� Minimum gaps are minimum clearance (or in case of interference fits, maximum interference)

Beginning Tolerance Stack-up Analysis

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� Its important to mentally shove all the features and parts in the

directions that will create the max or min gap (variable). This is to

allow your routes always pass through material and you don’t

want to jump over an air space unnecessarily in analysis

� You should position the features of the parts against each other so

that you will get extremes and make clear to you the correct path

and +ve v/s –ve designations for each number.

Beginning Tolerance Stack-up Analysis

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Finding Mean Dimensions

� Few Important Concepts of Tolerance Stack-up

Analysis:

– There is NO difference between equal, unequal or unilaterally

toleranced dimension.

– There is NO difference between a limit dimension and a plus

or minus toleranced dimension

– They all have extremes and they all have means. So, first thing

is to change any dimension to an equal bilateral toleranced

dimension

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Finding Mean Dimensions

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Finding Mean Dimensions

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Finding Mean Dimensions : Exercise

Convert following Dimensions to an equal bilateral toleranced dimensions

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Boundaries

Boundaries are generated by collective effects of

size and Geometric tolerances applied to

feature(s) and often referred to as simply inner

and outer boundaries

There are two types of boundaries

� Virtual Condition boundary (VCB)

� Resultant Condition Boundary (RCB)

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� FCFs that use m (MMC symbol), generate constantboundaries (VCB) for features under consideration and are calculated as:

– VCB for internal FOS such as hole = MMC Size Boundary –Geometric Tolerance value

– VCB for external FOS such as pin = MMC Size boundary + Geometric Tolerance

VC Boundaries are Constant and do not vary based upon actual mating size of the feature

Virtual Condition Boundaries (Refer ASME Y14.5M section

2.11)

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Virtual Condition Boundaries (Refer ASME Y14.5M

section 2.11)

� FCFs that use l (LMC symbol), generate constant

boundaries (VCB) for features under consideration and

are calculated as:

– VCB for internal FOS such as hole = LMC Size Boundary +

Geometric Tolerance value

– VCB for external FOS such as pin = LMC Size boundary -

Geometric Tolerance.

VC Boundaries are Constant and do not vary based upon actual

mating size of the feature

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Resultant Condition Boundaries (Refer ASME

Y14.5M section 2.11)

� RC Boundaries are non constant in nature and

are generated on opposite side of the virtual

conditions.

� When RFS (Regardless of Feature Size)

concept applies to FOS, they generate only

non-constant or RC boundaries.

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Case#1: Internal FOS controlled at MMC

Hole – MMC Concept

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Case#1: Calculating VC & RC boundaries

VCB for internal FOS (such as hole) controlled at MMC = MMC Size Boundary – GeometricTolerance value

VCB for external FOS (such as pin) controlled at MMC = MMC Size boundary + Geometric Tolerance value

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Case#1: Creating equal Bilateral Toleranced Dimension from VCB and RCB

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Case#2: Internal FOS controlled at LMC

Hole – LMC Concept

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Case#2: Calculating VC & RC boundaries

VCB for internal FOS (such as hole) controlled at LMC = LMC Size Boundary +Geometric Tolerance value

VCB for external FOS (such as pin) controlled at LMC = LMC Size boundary - GeometricTolerance value

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Case#2: Creating equal Bilateral Toleranced Dimension from VCB and RCB

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Case#3: Internal FOS controlled at RFS

Hole – RFS Concept

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Case#3: Calculating RC boundaries

Since it’s a RFS Callout, no virtual condition

boundaries exist and all boundaries are non-constant

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Case#3: Assumption about feature form in case of RFS callout

Only if the hole has a significant

depth, might this median line

curvature (out of straightness) be a consideration. For thin

parts, such as sheet metal, it is

probably not of concern in these

analyses. In fact many

designers would agree that a

banana shaped hole is not likely

to occur on most products.

Therefore we are ignoring

“axially out of straightness”

consideration from the

analyses.

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Case#3: Creating equal Bilateral Toleranced Dimension from Inner and Outer Boundaries

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Case#4: External FOS Controlled at MMC

Shaft – MMC Concept

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Case#4: Calculating VC & RC boundaries

VCB for internal FOS (such as hole) controlled at MMC = MMC Size Boundary – GeometricTolerance value

VCB for external FOS (such as pin) controlled at MMC = MMC Size boundary + Geometric Tolerance value

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Case#4: Creating equal Bilateral Toleranced Dimension from VCB and RCB

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Case#5: External FOS controlled at LMC

Shaft – LMC Concept

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Case#5: Calculating VC & RC boundaries

VCB for internal FOS (such as hole) controlled at LMC = LMC Size Boundary +Geometric Tolerance value

VCB for external FOS (such as pin) controlled at LMC = LMC Size boundary - GeometricTolerance value

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Case#5: Creating equal Bilateral Toleranced Dimension from VCB and RCB

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Case#6: External FOS controlled at RFS

Shaft – RFS Concept

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Case#6: Calculating RC boundaries

Inner Boundary

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Case#6: Assumption about feature form in case of RFS callout

In case of RFS Callout, one may want to consider

additional deviation arising ‘out of form’ to

determine absolute worst case inner boundary.

This applies only if shaft length is significant. For

very shirt shafts / pins, it is probably not of

concern in the analysis.

Therefore we are ignoring ‘out-of-straightness’

consideration from our analysis. If your product

runs a risk of banana shaped shafts, you may

wish to consider illustration on the left in your

calculations.

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Case#6: Creating equal Bilateral Toleranced Dimension from Inner and Outer Boundaries

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Formulae to Remember…

For External FOS controlled at MMC / LMC:

VCB at MMC (OB) = MMC Size boundary + Geometric Tolerance value at MMC

VCB at LMC (IB) = LMC Size boundary - Geometric Tolerance value at LMC

For Internal FOS controlled at MMC / LMC:

VCB at MMC (IB) = MMC Size Boundary – Geometric Tolerance value at MMC

VCB at LMC (OB) = LMC Size Boundary + Geometric Tolerance value at LMC

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Finding Inner & Outer Boundaries : Exercise

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Session #2: Analyzing a Box Assembly

� Objectives:

To determine min and max gap for a simple

eleven parts assembly

� Perform the calculations

� Create a Loop Analysis Diagram

� Create a Number Chart

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Box Assembly

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Box Assembly: Part #1

� Cavity in part#1 has limit dimensions (392.43 –

384.81); so we need to convert these to mean

with equal bilateral toleranced dimensions …

• Add limit dimensions : 392.43 + 384.81=777.24

• Find Mean dimension: 777.24/2=388.62

• Find total tolerance by subtracting limit dimensions: 392.43 –

384.81=7.62

• Find equal bilateral tolerance=7.62/2=3.81

• Finally express limit dimensions as equal bilaterally toleranceddimension as: 388.62`3.81

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Box Assembly: Part #2- #11

parts MMC … (1)

parts LMC … (2)

Add (1) and (2) …(3)

Subtract (1) and (2)…(4)

Half of (3) …(5)

Half of (4) …(6)

(5)

(6)

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Box Assembly: Part #1 & Part #2- #11 put together

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Box Assembly : Loop Analysis Diagram

�Loop Diagram begins by showing Gap to be

calculated at the top

�So, loop diagram begins at Part#11 (plate)

and progresses downward constantly through

material until it reaches at the last plate at the

bottom of an assembly (ie. Part#2 or plate#2)

�The sum of all these –ve mean dimensions,

which run from top to bottom is 381and has

total tolerance of `3.81

�The loop then reverses and progresses up

through cavity (ie. Part #1).

�This portion of the loop is +ve since it

progresses from bottom to top

�The logic of +ve and –ve is simple… material

removes airspace (therefore –ve) and cavity

which lacks material adds to airspace (therefore +ve)

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Box Assembly : Loop Analysis Diagram

�The mean dimension of cavity is 388.62 and has a total tolerance of `3.81

�So, in numbers chart, we add means: (+)388.62 + ( -

)381.00 = 7.62 …(1)

�If this number is –ve, it would have proven that even

mean sizes of parts, when produced result in

interference. Now that sum is +ve, we can proceed…

�Next step is to add, charted plus or minus tolerances : 3.81+3.8 1 = 7.62 …(2)

�Next step is to calculate min and max gaps (airspace or interference):

�Mean dimensions difference + sum of tolerances = (1) + (2)= (+)7 .62+(+)7.62=+15.24 (max gap)

�Mean dimensions difference - sum of tolerances = (1) - (2)= (+)7.62-(+)7.62=0 (min gap)

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From situations such as this, it is easier to simply calculate t he MMC of the

cavity and the collective MMCs of the plates and subtract them to get

minimum gap.

Box Assembly : Alternate Method to calculate min / max gap

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Box Assembly : Loop Analysis Diagram

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Session #2: Exercises

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� Objectives:

– Using Loop Analysis Technique; determine Max and Min gap

in Horizontal and Vertical Directions

– Determine proper start and End points for stack-ups

– Graph the numbers calculated into Loop Diagram

Session #3: Loop Analysis for Features of Size (FOS)

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Analyzing FOS: Problem Description

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Analyzing FOS: Charts to be used…

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Analyzing FOS: Steps Involved (HorizontalDirection)

� Convert horizontal limit dimensions of part #1 to equal bilatera l toleranced dimension. (26.615`0.405)

� Convert horizontal limit dimensions of part #2 to equal bilatera l toleranced dimension. (25.705`0.105)

� Graph the loop from left-to-right through material, using appropriate signs (+ve / -ve) (- 25.705 and + 26.615)

� Add these mean dimensions (don’t forget signs) to get difference between mean dimensions ((-) 25.705 + (+) 26.615)=+0.910

� Add plus and minus tolerances for part#1 and part#2 to get total plus and minus tolerance (0.105+0.405)=0.510

� Max gap in horizontal direction is = sum of difference between mean dimensions and total of plus and minus tolerances (0.910+0.510=1.42)

� Min gap in horizontal direction is = difference (subtraction) of differencebetween mean dimensions and total of plus and minus tolerances (0.910-0.510=0.4)

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Analyzing FOS: Steps Involved (VerticalDirection)

� Convert vertical limit dimensions of part #1 to equal bilateral toleranceddimension. (26.615`0.405)

� Convert vertical limit dimensions of part #2 to equal bilateral toleranceddimension. (24.390`0.610)

� Graph the loop from top-to-bottom through material, using appropriate signs (+ve / -ve) (- 24.390 and + 26.615)

� Add these mean dimensions (don’t forget signs) to get difference between mean dimensions ((-) 24.390 + (+) 26.615)=+2.225

� Add plus and minus tolerances for part#1 and part#2 to get total plus and minus tolerance (0.610+0.405)=1.015

� Max gap in vertical direction is = sum of difference between mean dimensions and total of plus and minus tolerances (2.225+ 1.015)=3.24

� Min gap in vertical direction is = difference (subtraction) of differencebetween mean dimensions and total of plus and minus tolerances (2.225-1.015)=1.21

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Analyzing FOS: Easier Method using MMC and LMC

Calculating min and max gaps may be easier as discussed before ( slide #48); by

subtracting the MMCs for minimum gaps and LMCs for maximum gaps as

shown below:

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Analyzing FOS: Charts and Loops with dimensions…

Part #1, horizontal

direction

Part #2, horizontal

direction

Part #1, vertical

direction

Part #2, vertical

direction

Material side (-ve)

Part #2 thickness

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Session #3: Exercise

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Session #4: Analysis of an assembly with Plus and Minus tolerancing

� Objectives:

� Calculate the airspaces and interferences for a plus and

minus toleranced assembly

� Performing multiple loop analyses on an assembly

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Assembly with plus and minus tolerances : Problem Description

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Assembly with plus and minus tolerances : Charts to be used