Introduction

Before Geometric Dimensioning \( \& \) Tolerancing(GD\( \& \)T), engineers used standard coordinate and plus/minus systems to convey ideas through drawings. This often left the fabrication of the product to the engineers interpretation. GD\( \& \)T can relate tolerances to their respective datum planes to eliminate ambiguity across engineering drawings.

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Surface Roughness

In the design process, the engineer must account for the surface textures of their parts. This can be for aesthetics of a part, considering safety for walkers, mechanical and physical properties, assembly of parts can be affected by their surface contact points, and electrical connections are made better on smoother surfaces.

How Can We Define a Surface Texture?

We can break up the description of surface texture into 4 categories.

• Roughness: Small, finer-spaced deviations from nominal surface.
• Waviness: Larger deviations due to larger spacing, often occurring from work deflections, vibrations, etc.
• Lay: Direction or pattern of surface texture
• Flaws: Localized irregularities likes cracks, scratches, or divots

Surface Roughness Value

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When considering Surface Roughness, we can base our values from using a stylus across a surface to detect heights y across the surface with equal distance apart from each point of measurement. We can insert our values of height \( \& \) number of measurements, n, into the following equations:

• Arithmetic Mean Equation: Gives average roughness between all the heights of the surface, less sensitive to peaks
$$ R_a = \frac{|y_a|+|y_b|+|y_c|+\cdots+|y_n|}{n} = \frac{1}{n}\sum_{i=1}^{n}|y_i| = \frac{1}{L}\int_{0}^{L}|y|\,dx $$
• Root-Mean-Square Equation: Like Arithmetic, but more sensitive to peaks \( \& \) valleys in the surface
$$ R_q = \sqrt{\frac{y_a^2 + y_b^2 + y_c^2 + \cdots + y_n^2}{n}} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}y_i^2} = \sqrt{\frac{1}{L}\int_{0}^{L}y^2\,dx} $$

However, a problem that arises is the equations for roughness above which is the waviness may get included and interfere with the calculation for roughness. To combat this it is recommended that you select a cut-off length to filter the waviness from the roughness calculation. In order to do this, a sampling distance shorter than the waviness deviations can give better results for roughness.

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Engineering Drawings Terminology

Standardized symbols for engineering drawings can be used for describing surface texture if material removal is permitted or not. They are displayed as the following

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There is also patterns that can be machined into the surfaces dependent of their drawings. The symbol will most commonly be placed next to the symbol denoting a surface. They can be any of the following:

• "=" - Tooling is parallel to projected surface drawing
• "\( \perp \)" - Tooling is perpendicular to projected surface
• "\( \times \)" - Cross hatched pattern on indicated surface
• "M" - Non-directional or multi-intersection on indicated surface
• "C" - Concentric circles roughly centered
• "R" - Radiating shape, roughly passing through the centroid in each pass

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Geometric Dimensioning & Tolerancing (GD&T)

In engineering drawings, it is critical to use GD\( \& \)T ensure one interpretation of engineering drawings

Definitions

• Dimension: A numerical value in appropriate units to define size or geometric characteristic
• Tolerance: The total amount by which a specific dimension is allowed to vary

Requirements of Drawings

1. Drawing must not be subject to more than one interpretation
2. Dimensions should suit the mating relationship between parts
3. Perpendicularity is assumed when no angle is expressed.

Geometric Characteristic Symbols Chart

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How to Read GD&T

When reading GD&T engineering drawings, there are 4 main features to notice to interpret the drawing correctly

• Geometric Characteristic: Characteristic symbol used to describe and determine the shape of the feature
• Tolerance: The total amount by which a specific dimension is allowed to vary
• Datum: Where part is first located using surface A, then to surface B, then to surface C. This will fully constrain the part, and prevent translational movement in the X, Y, Z-directions, and as well as as rotational movement on those respective axes(pitch, roll, yaw)
• Material Conditions: Conditions in which features can respond to changes in size from one side of a connection to another. They are the following:
• Maximum Material Condition (MMC): Where the part weighs the most. This is the case where largest shaft or pin size can fit inside, or the smallest allowable hole size
• Least Material Condition (LMC): Where the part weighs the least. This is the case where it fits the smallest shaft or pin size, or the largest allowable hole size
• Regardless of Feature Size (RFS): Tolerance stays the same in every case the manufacturer follows the drawing, (i.e., no possible bonus tolerance)

Once all have been given on the drawing, first take note of what the dimensions are on the drawing. Then look towards the control frame, see what geometric characteristic symbol the surface is trying to describe. The numerical value will be the geometric tolerance of the surface with the given material condition, relative to the datums in their respective order.

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Composite Feature Control Frames

Sometimes when making the drawings of our parts, we will want to set higher tolerances on specific features of a product. This method of tolerance will display as the following in engineering drawings:

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The single shared symbol feature will have the upper segment that controls the feature as normal as a general tolerance, and the lower segment refines the feature with more specified tolerances

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Bonus Tolerance

In some instances, bonus tolerances can occur when the hole size is the MMC size if specified in the drawings. If the hole size is bigger than the MMC size, we get a bonus tolerance equal to the difference between the MMC size and the actual size. This ranges from the MMC size to the LMC size.

Goals of GD&T:

• To set the standard of maximum and minimum limits of the dimensions of length and angle
• Dimensional control over product, as exact dimensions in manufacturing is not achievable
• Set tolerance as large as possible to minimize manufacturing cost while keeping the tolerance small enough to ensure safe operation and interchangeability

Quality Assurance

Definitions

• Quality Assurance: Total effort by a manufacturer to ensure its products are abide by a set of specifications and standards, they may include:
• Surface Finish
• Dimensional Tolerances
• Composition
• Aesthetics
• Mechanical, physical, or chemical properties of materials
• Measurement: Procedure in which an unknown quantity is compared to a known standard or reference value, using a system of units. Devices to measure with can be a micrometer, caliper, protractor, etc.

Accuracy and Precision

The mission of quality control is to ensure product is verifiable to function as intended, while still having small deviation in part size and shape due to tolerances.

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• Accuracy: The degree to which a measured value agrees with the true value of the quantity of interest
• Precision: The degree of repeatability in the measurement trial process

Approaches to Quality Control

For methods to improve quality of a product, there can be two ways to make progress:

• Inspection and Part Rework: Typically during this process, inspection is made and rejection of parts that are not abiding by standards.

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• Continual Process Improvement:

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This process tends to be more costing, but this will improve product processing, decreasing the percentage of parts being rejected over time, making this approach much better overall in the long-run

Gaussian Distributions:

Gaussian distributions are used to depict how well the product are manufactured by orders of variance in the part. We can use the distribution to determine the Variance and Tolerance of a system.

A Gaussian contains the following

• Mean: \( \mu \)
• Standard Deviation: \( \sigma \)
• Variance: \( \sigma^2 \)
• Tolerance:
• \( \sigma \)=68.2
• \( \sigma \)=95.4
• \( \sigma \)=99.7

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We then can use these values to calculate the following: