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In the previous post (Standards of limits and fits for mating parts), we defined the core terms of the ISO 286 standard.
Now, we translate that theory into real-world Precision Metrology calculations. While manual math is good for understanding the "why," modern manufacturing relies on Tolerance Analysis Software to prevent costly scrap in CNC Machining.
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Automated Calculation Logic
Below is an example of how logic is structured in engineering spreadsheets or professional Quality Control (QC) software to determine upper and lower deviations.
Step-by-Step Manual Calculation
Let's verify the software's logic by calculating the deviations manually.
Example: Calculate the deviations for a shaft with a diameter of 40 mm and a tolerance class of g6 (40g6).
Step 1: Determine the Geometric Mean (D)
First, we find the basic size range. For a 40 mm shaft, the range is "Over 30 up to 50 mm" (D_min = 30, D_max = 50).
Calculate the Geometric Mean (D):
D = SQRT(30 × 50) = 38.73 mm
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Step 2: Calculate Standard Tolerance Unit (i)
Using the ISO formula for grades IT5–IT16:
i = 0.001 × [ 0.45 × (38.73)^(1/3) + 0.001 × 38.73 ]
i = 0.00156 mm
Step 3: Calculate IT Grade Tolerance
For grade IT6, the standard multiplier is 10i.
Tolerance = 10 × 0.00156 = 0.0156 mm (15.6 microns)
Step 4: Determine Fundamental Deviation
For a "g" shaft, the fundamental deviation is the Upper Deviation (es).
Coefficients for "g" shaft: a = 0, b = -2.5, exponent = 0.34
Formula: Fundamental Deviation = (b × D^0.34) / 1000
Result = (-2.5 × 38.73^0.34) / 1000 = -0.009 mm
Final Results for 40g6:
Upper Deviation (es) = -0.009 mm
Lower Deviation (ei) = Upper Deviation - IT Tolerance
ei = -0.009 - 0.0156 = -0.025 mm (approx)
Upper Deviation (es) = -0.009 mm
Lower Deviation (ei) = Upper Deviation - IT Tolerance
ei = -0.009 - 0.0156 = -0.025 mm (approx)
Using Digital Tools for Validation
In a production environment, engineers use Computational Engineering Tools to avoid human error.
1. Input the Basic Size:
Entering "40" automatically selects the correct geometric mean range.
2. Select Tolerance Class:
Selecting "g" and "6" triggers the deviation lookup logic we calculated above.
Why Use Professional Software?
While Excel sheets are great for education, professional shops rely on GD&T (Geometric Dimensioning and Tolerancing) Software and integrated CAD/CAM solutions. These tools automatically validate fits against ISO 286 standards, ensuring that a precision shaft will always fit its mating bearing, reducing the risk of expensive rework.
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