0: Intro & Tolerancing

01-18: Course introduction #

Overview #

This class focuses on three main components – manufacturing processes, dimensional tolerances, and design communication – and how they interact with one another.

The class is made up of 9 modules:

  1. Fundamentals
  2. Subtractive manufacturing processes
  3. Additive manufacturing processes
  4. Forming processes
  5. Joining processes
  6. Graphical visualization techniques
  7. Metrology: measuring manufactured objects
  8. Geometric dimensioning and tolerancing
  9. The future of manufacturing

Why manufacturing? #

  • In 2018, U.S. manufacturing accounted for 11.6% of the U.S. economy, 18.2% of global manufacturing output, and 8.2% of the U.S. workforce – source
  • Manufacturing output is growing, and is returning to the U.S.; output increased >30% between end of 2008 and 2014
  • 67% of U.S. R&D is funded by industry
  • Even when production is offshore, design is often done here anyway
  • Automation is increasing, yet there is a shortage of skilled (human) talent
  • Even if you don’t want to go into manufacturing industry, research and academia still require manufacturing knowledge
  • Even if the process if outsourced, design is still done in the US. To design well, you have to have a base-level understanding of manufacturing

Manufacturing output and employment are rising

Many companies have regionalized their supply chains since the pandemic

Processes #

In this class we will consider multiple families of processes: Processes

  • This is a rapidly moving field that is always adapting
  • This class should give you a top level overview so you can evaluate novel methods

Materials #

In this class we will consider multiple families of materials: Materials

  • Materials choices influence performance
    • For example, consider the progress of the plane:
    • In 1903 the right brothers low-density wood with steel wire and silk
    • In 1935 the Douglas DC3 used aluminum alloy (since it became feasible to produce and manipulate)
    • Now the 2010 Boeing 787 Dreamliner is made up of 50 wt% composites 20 wt% aluminum 15 wt% titanium 20% lower fuel consumption per passenger mile
      • Composite materials are two(+) materials combined together to get best of both worlds, in aviation typically stiff/strong carbon fibers embed in tough/fatigue-resistant polymers.
  • Materials choices influence market size Cams
    • There isn’t always a best material; different materials fit different markets/needs
    • Opposite side of the coin: There may be multiple valid material choices for a particular function

Tolerance #

  • Tolerancing is a formal way of specifying limits on the amount of dimensional variability allowable in manufactured parts
    • We need a range because measurements will never be 100% precise; we need to define an acceptable range
    • Some sources of variation
      • Human operator changes and/or errors
      • Tool wear
      • Environmental changes (temperature, humidity leads to tiny expansions / contractions)
      • Input material variability
      • Measurement error
  • Affordable mass-production relies on interchangeability of parts
    • When mating parts of given designs, it should not matter which specific parts
  • Therefore part dimensions must be consistent
    • But no manufacturing process is perfectly consistent
  • If you don’t understand the process of manufacturing and the capabilities of tools, then you will won’t know how to create manufacturable designs
  • Tighter tolerances (closer tolerance limits) are generally more expensive to achieve
  • The solid green line shows an ideal process
  • The dotted green line shows the impact of an error shifting the distribution, shifting the tails to approach the tolerance upper / lower bound
  • The red line shows a unsuitable process (even if it’s calibrated accurately, the poor precision causes high variance that it’s not really feasible; however, if outside of the limits an additive (or less common subtractive) could be used to )

How E29 integrates manufacturing and tolerancing #

  • Tighter tolerances are more expensive
  • The physics of a process determine how tight a tolerance is achievable and how much it costs
  • Therefore we need to understand how manufacturing processes work in order to:
    • Select a suitable process for the application
    • Specify reasonable tolerances
    • Geometric Dimensioning and Tolerancing: a graphical language for specifying tolerances robustly

Design Communication #

  • Important to effectively describe your ideas and designs graphically
    • Persuade – we need to be able to show are perspective
    • Instruct – we need an agreed an unambiguous way to communicate
    • Document – we need to convey how to construct our final design
    • Seek feedback – we need to ensure everyone is on the same page
  • Drawings can be 2D or 3D representations
    • Interpreting 2D drawings made by others
    • Creating 2D “working drawings” with unambiguous instructions
  • Design communication is not only graphical
    • Oral, written
    • Manufacturing relies on teams
    • Teaming activities

01-20: Fundamentals of Tolerancing #

Basic tolerance formats #

  1. Unilateral
    • e.g. Inches: .$.500^{+0.005}_{-0.000}$, Metric: .$35^{+0.05}_0$ (notice sigfig notation)
  2. Bilateral
    • Most common; start with nominal then you have some tolerance bounds above and below
    • Equal or unequal deviations from nominal dimension
    • Same number of decimal places for upper and lower limits
    • e.g. Inches: .$.500 \pm .005$ or .$.500^{+0.005}_{-.010}$, Metric: .$35 \pm 0.05$ or .$35^{+0.05} _{-0.10}$
  3. Limit
    • Given only bounds, not the nominal value
    • e.g. Inches: .$.250, .248$, Metric: .$35.05, 35.00$

Tolerance buildup #

  • In the real world we have error, so the way we define dimensions have an impact
  • Best dimensions to label depend on function
    • That is, dimensioning should be done intentionally such that critical distances result in minimal error,
    • e.g.suppose distance between .$X$ and .$Y$ is critical

Side Projection

  • Chain is bad since the potential (and often times real world) maximum error is large
    • The errors compound since dimensions are in reference to other dimensions that may will contain error.
    • The more dimensions chained, the greater the possible error
  • Baseline is better – every feature references a single base.
    • However the worst case is still significant
    • .$X$ may be off by .$\pm .05$ and .$Y$ may be off .$\mp 05$, compounding to .$\pm 0.10$!
  • Direct is ideal
    • Depends on which dimensions are critical (that is, .$X, Y$)

Normal cumulative distribution function #

  • Tighter tolerances (closer tolerance limits) are generally more expensive to achieve
  • The physics of the process used determines the curve’s characteristics
    • .$\sigma$ is the stdiv (width) of this density
  • .$\mu = x_0$ is the target (average) value we give
  • This probability density characterizes how this function is distributed and the chance a given range of values occur
    • The area under the curve in a given range is the probability the value falls within that range
    • Single values, i.e .$x_0$, have a 0% probability. We can only calculate ranges because this is a density function.

Distribution Probability density, e.g. given by Gaussian/Normal probability density function: $$p(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{(x-x_0)^2}{2\sigma^2}}$$

  • Why do we care about statistics?
    • We want to look at a process, look at tolerances, and figure out whether it’s worth to manufacture using this process
    • If you know the distribution of a process, you can work out the probability a given part satisfies spec limits.
  • There is no easy, exact analytical way to integrate the normal probability density function.
    • The probability that a randomly chosen member of a normally distributed population has a value .$\leq x$ is $$\int_{-\infty}^x p(x)\ dx = P(x) = Z\bigg(\frac{x-\mu}{\sigma}\bigg) = \frac{1}{2}\bigg[1 + \text{erf}\bigg(\frac{x-\mu}{\sigma\sqrt{2}}\bigg)\bigg]$$
Example probability of part lying between within spec limits
Distribution Example

Process capability and tolerancing #

  • Sigma, .$\sigma$, is the standard deviation of dimensions actually produced by a process
  • Six sigma processes

    Six Sigma (6σ) is a set of techniques and tools for process improvement. […] A six sigma process is one in which 99.99966% of all opportunities to produce some feature of a part are statistically expected to be free of defects.

    • Specification limits are .$12\sigma$ apart. Here, 2 parts per billion lie outside specification limits if process is ‘in control’ (i.e. if mean output of process is centered between specification limits)
    • Arose because the cost of manufacturing, specifically the process that creates an error, has a cost. This cost can grow very large, very quickly, when mass-manufacturing.
      • You’re best off spending money improving the process so the distribution gets tighter
      • The alternative is either (1) accepting errors (resulting in faulty products) or (2) testing all components to ensure they are ‘good’ and tossing out the bad ones
    • Process capability: .$C_p = \frac{\text{USL - LSL}}{6\sigma}$

Classes of Fit #

  • Tolerances should be…
    • Not too tight: tight tolerances are expensive
    • Not too loose: otherwise function is compromised


  1. Clearance fit: designed with space left between two components
    • e.g. a shaft with a bearing need to have some give / free space
  2. Interference (push) fit: designed to be touching
    • You may want interference because you want the friction between the components; you want the two pieces to not move/rotate/etc
    • How? Elastic or even plastic deformation
    • e.g. two pieces may need to fit tightly with friction as to prevent vibrations
    • Expansion fit:
      • If there are large forces/torques acting on these two components so you want them very tight
      • e.g. you may temporarily expand one component (e.x. with heat) to fit on/around the other, then it will shrink down
    • Shrink fit:
      • Same as expansion, but using some cooling process (e.x. liquid nitrogen)
      • Why do this over heat?
        • It’s typically more expensive to cool down
        • The material may deform / weaken – e.g. steel will be degraded if heated up
  3. Transition fit: complete interchangeability is compromised to allow looser tolerance on individual components.
    • If fit type is not critical.
    • But even then, why not choose one or the other? Because you don’t want a large gap and the materials/parts cannot withstand the force needed to assemble them with an interference fit.
    • The pieces are just for alignment – think Ikea assembly pegs; they’re just to align components.
    • It’s easier to manufacture these parts

Snap fits #


  • Involves temporary elastic deflection which enables parts to interlock, e.g. involving bending of one component
  • Done often with molded parts
  • Tends to involve Cantilever (e.g. casings), Annular (e.g. pen lids, take-out soup container lids)
  • Designed to be assembled once, and typically not disassembled (multiple times) – irreversible.
  • Relatively simple: you don’t need screws/glues/etc. – useful for rapid prototyping since you don’t have to consider fasteners
  • Takes advantage of the fact that the material has some elasticity
    • You need to stay within the elasticity limits of the material
    • Most 3D plastics have ’enough’ give
    • You (generally) want to design such that the stress is from bending, not stretching
  • More, additional, extra, readings

Terminology Definitions #

Don’t stress about memorizing these !

Clearance and interference fits

  • Maximum material condition (MMC): The greatest allowable amount of material left on the part (max size for a shaft; min size for a hole)
  • Minimum/least material condition (LMC): The least allowable amount of material left on the part (min size for a shaft; max size for a hole)
    • Important with MMC since they tell us how much they’re able to ‘slosh around’
  • Basic size: Exact theoretical size from which limits are derived
    • Different form nominal since basic refers to the standard table which gives respective upper and lower bounds (MMC and LMCs)
    • Hole basis: Basic size is minimum size of hole
    • Shaft basis: Basic size is maximum size of shaft – used when many components need to fit on to one shaft.
    • Basic size could be chosen to be in-between hole and shaft basis
  • Tolerance: Allowable variation of one particular dimension
  • Fundamental deviation: Difference between basic size and the closer of the MMC and LMC
  • Allowance: Difference between maximum material conditions of the two components

Types of fit #

  • These types are created by ANSI: American National Standards Institute
  • Exact values are tabulated in many source

  • RC: Running and sliding clearance fits
    • Nine categories:
      • RC1: Close sliding: assemble without perceptible “play” (e.g. watches)
        • Less than a 1/1000".
        • Basically impossible for air, let alone liquids, through.
      • RC2: Sliding fits: seize with small temperature changes (e.g. )
      • RC3: Precision running: not suitable for appreciable temperature differences
      • RC4: Close running: moderate surface speeds and pressures
      • RC5/6: Medium running: higher speed/pressure
      • RC7: Free running: where accuracy not essential and/or temperature variations large
      • RC8/9: Loose running
    • Go for lower if you want minimal vibration/gaps – no perceivable play.
    • Has drawbacks:
      • The less clearance, the easier it is to seize up – especially if two components are touching and made up of different materials (different expansion/contraction rates).
      • Susceptible to dust, you would have to seal the machine or use it in clean conditions.
    • If you go less precise, you don’t need to go slow, cheaper operator costs, cheaper tooling
    • RC Chart
      RC fits – from Machinery’s Handbook, Industrial Press
    • RC Table
  • LC: Locational clearance fits
    • Normally stationary, but freely assembled/disassembled
    • Used when you need clearance to dis able and clean
    • LC Chart
      Classes of LC fit
  • LT: Location transition fits
    • Accuracy of location important
    • Small amount of clearance or interference OK
    • e.g. ikea furniture pegs
  • LN: Locational interference
    • When you need friction
    • Accuracy of location is critical

Other classes of fit

  • FN: Force fits
    • When you need to hold a load (typically uses temporary heating)
    • Designed to transmit frictional loads from one part to another
Example: Which type of fit?

Processes, tolerances, and surface quality #

  • How do we relate physical processes and tools to these values?

From MF Ashby, Materials Selection in Mechanical Design

Roughness #

  • How do we define roughness? You may use tool that uses a tiny needle to ‘scan’ the surface, measuring deflections as you go

From MF Ashby, Materials Selection in Mechanical Design

  • RMS roughness: root mean square of deviations over the measured surface length
    • i.e.: .$R^2 = L^{-1} \int_0^L y^2\ dx$
    • Usually, tolerance, .$T$, lies between 5R and 1000R
  • Generally, if you go high rotation speed and slow translational speed you get less rough surfaces
    RMS Roughness Example