This is to illustrate how to use tolerance analysis as a tool to ensure that a kitchen appliance molding design meets its mass production requirements and customer expectations for component assembly, product appearance, and product performance.

**The Benefits of Tolerance Analysis Before mass production**

- Verifies a design will achieve the desired level of quality
- Ensures proper product performance and reliability
- Saves manufacturing costs because products successfully assemble with fewer defects
- Reduces a product’s Field Failure Rate (FFR) which reduces warranty costs

**Tolerance Analysis Methods**

Linear stack up analysis

- Worst case stacking
- Statistical stacking – RSS (Root Sum of Squares)

Non linear stack up analysis

- 3-D tolerance analysis software to analyze variation in 3-D solids-based CAD assembly models. which can handle 3-D stack-ups, even with GD&T (Geometric Dimensioning and Tolerancing) defined features.
- Monte Carlo Simulation (good for stack-ups involving dimensions with non-normal distributions)

Suzubin focus on linear stack up analysis.

**Tolerance Analysis Process**

- Establish the design requirement
- Create loop diagram
- Convert tolerance to be bilateral symmetric
- Determine method of tolerance analysis
- Apply the TA method to assess gap variation
- Adjust design to meet requirement

**1, Establish the Design Requirement**

Appearance requirements

- Visual: consistency on the product’s visible styling groove gap
- Ensure perception of quality: avoid rattling noises due to parts that are too loose

Manufacturing Requirements

- Interchangeability (successful assembly) of parts coming from multiple tools across multiple suppliers
- Avoid over-specification: design for tolerances that allow purchase of components at low cost

Functional requirements

- Electrical: ensure that
*intended*electrical connections always occur when they should, and*unintended*electrical shorts never occur - Mechanical movement: proper functioning of sliding parts, housing covers that remove/assemble with ease but are not too loose
- Reliability: assembly variation does not stress a particular component beyond its physical limit

Let’s examine the design requirement for the “blocks in slot” assembly problem. Notice below how design requirements are first described in terms of fit, function or appearance; *then those requirements are translated into mathematical formulas which bound the Assembly GAP:*

**2, Creating the Loop Diagram**

Create loop diagram:

- Assigned a coordinate system, and sketched it beside our diagram.
- Visualize loop segments as “neighboring” dimensions; sometimes it helps to visualize moving components so they touch each other.
- Established loop segments. Express the loop segment dimensions as vectors.
- We also sketched the GAP segment as a resultant vector with a direction.

Visualize parts as touching:

Moving part closer together to establish solid geometry connections, eliminating all gaps (except for the assembly GAP, which is the gap-under-study)

Create each successive loop vector. A loop’s segments must “walk” from one side of the assembly gap, through solid part dimensions that “touch” each other, until reaching the opposite side of the assembly gap.

Calculate nominal assembly gap.

*R*** NomGap** = the nominal value of the gap.

**“+” sign = clearance, “-” sign = interference.**

** n** = the number of independent dimensions in stack.

*D*** i** = the nominal value of the

*i*th vector dimension in the loop diagram.

*V*** i **= the

*i*th

*signed*unit vector corresponding to the

*i*th nominal loop dimension

**3, Conversion to Symmetric ****Bilaterial**** Tolerances**

- From a design perspective, all of the dimensioning methods above are equivalent.
- Uses RSS formulas, requiring input of bilateral symmetric tolerances.

**4, Determine method of tolerance analysis**

*Most commonly used tolerance models are:*

Worst Case (WC) model

- Verifies 100 % performance
- The simplest and most conservative approach
- Used when number of loop segments is
*less than four* - Also can be used for unusually low volume production

Root Sum of Squares (RSS) model

- Statistical formula-based approach, which requires that loop segments have normal distributions, and are independent of each other
- Used when there are
*four or more*tolerance dimensions in the tolerance loop

Which method to use for below blocks example?

Check: Do the loop segments have normal distributions?

Answer: Yes, most manufacturing processes (machining, injection molding …etc.) produce component populations with normal distributions.

Check: How many loop segments are there?

Answer: Four (Note: the “GAP” doesn’t count as a loop segment)

Use **RSS** because this assembly below has four, and each has a normal distribution.

Which method to use for our blocks example?

Check: Do the loop segments have normal distributions?

Answer: Yes

Check: How many loop segments are there?

Answer: Three

Use worst case because this assembly below has less than four dimensions, and each has a normal distribution

**5, Apply the TA Method to assess GAP variation (WC)**

Worst Case tolerance of assembly gap is simply the sum of individual tolerance.

*T**assy* = maximum expected variation (equal bilateral) of the gap.

*n* = the number of independent dimensions in the stack.

*d**i* = symmetrical bilateral tolerance of the *i*th dimension in the stack.

Use the following formula to calculate RSS.

*S**assy* = estimated standard deviation of the Assembly GAP.

*tol* = the bilateral tolerance for each dimension.

*Cp* = inherent process capability associated for each dimension’s tolerance and manufacturing process.

*n* = number of independent dimensions in the stack

Understanding how Cp level affects true variation.

Note: “B” and “C” have different tolerances and different Cp, but they actually have the same level of variation

The RSS formula acts to “sum” these component standard deviations to estimate the assembly distribution.

Use below formula to calculating the range of the assembly gap.

**Gap**assy **USL**** = ****Gap**Nomimal **+** **T**assy = 0.3 + 0.228 = + 0.528 mm

**Gap**assy **LSL**** = ****Gap**Nomimal **–** **T**assy = 0.3 – 0.228 = + 0.072 mm

**Gap**assy = 0.3 ± 0.228 = 0.072 mm ~ 0.528 mm

**“+” sign = clearance, “-” sign = interference.**

**6, Adjust design to solve analysis**

If assembly gap not meet design requirement, we need adjust design to meet requirement.

Example:

*Design requirement is gap>0, and gap<0.5mm.*

*If assembly gap <0, assembly interference. Solve this problem by adjust design:*

–* Increase block “A” length.*

–* Reduce block “B”/”C”/”D” length.*

–* Adjust tolerance definition.*

–

*If assembly gap >0.5mm, assembly loose. Solve this problem by adjust design:*

–* Reduce block “A” length.*

–* Increase block “B”/”C”/”D” length.*

–* Adjust tolerance definition.*