15.1 Limits, Fits, Tolerances, Gauges & Surface Metrology

Limits, Fits, Tolerances, Gauges & Surface Metrology

In heavy engineering and mining machinery manufacturing—such as the massive draglines, continuous miners, and longwall shearers deployed by Coal India Limited—interchangeability of mating components is paramount. Interchangeability ensures that any one component selected at random from a batch will assemble and function correctly with a mating component without custom fitting or secondary machining.

---

1. Terminology of Limits and Fits (IS 919 / ISO System)

The Indian Standard IS 919 (aligned with ISO 286) establishes a standardized framework for defining linear dimensional limits, tolerances, and mating relationships between cylindrical features (holes and shafts).

               +-------------------------------------------+  <-- Upper Limit of Hole (ES)
               |                                           |
               |               HOLE TOLERANCE (Th)         |
  Hole Size    |                                           |  <-- Lower Limit of Hole (EI)
               +-------------------------------------------+
============================================================= <-- ZERO LINE (Basic Size)
               +-------------------------------------------+  <-- Upper Limit of Shaft (es)
               |                                           |
  Shaft Size   |               SHAFT TOLERANCE (Ts)        |
               |                                           |  <-- Lower Limit of Shaft (ei)
               +-------------------------------------------+

Core Definitions

1. Basic Size ($D$): The exact theoretical size from which limits of size are derived by the application of deviations.
2. Actual Size: The size of a manufactured component obtained by direct measurement.
3. Limits of Size: The two extreme permissible dimensions between which the actual size must lie:
   • Upper Limit ($UL$): Maximum permissible dimension.
   • Lower Limit ($LL$): Minimum permissible dimension.
4. Tolerance ($T$): The total permissible variation in a dimension: $$\text{Tolerance} = UL - LL = \text{Upper Deviation} - \text{Lower Deviation}$$
5. Zero Line: A graphical reference line representing the basic size. Positive deviations lie above the zero line; negative deviations lie below it.
6. Deviations:
   • Upper Deviation ($ES$ for Hole, $es$ for Shaft): Algebraic difference between upper limit and basic size: $$ES = UL_{\text{hole}} - D_{\text{basic}}, \quad es = UL_{\text{shaft}} - D_{\text{basic}}$$
   • Lower Deviation ($EI$ for Hole, $ei$ for Shaft): Algebraic difference between lower limit and basic size: $$EI = LL_{\text{hole}} - D_{\text{basic}}, \quad ei = LL_{\text{shaft}} - D_{\text{basic}}$$
   • Fundamental Deviation: The deviation closest to the zero line that defines the position of the tolerance zone relative to the basic size.

---

2. Standard Tolerance Grades and Fundamental Tolerance Unit ($i$)

The ISO/IS system specifies 18 standard tolerance grades: $\text{IT01}, \text{IT0}, \text{IT1}, \text{IT2}, \dots, \text{IT16}$.

• Grades $\text{IT01}$ to $\text{IT4}$ are for high-precision master gauges.
• Grades $\text{IT5}$ to $\text{IT7}$ are for precision fits and fine manufacturing (grinding, broaching, reaming).
• Grades $\text{IT8}$ to $\text{IT11}$ are for general engineering production (turning, milling, drilling).
• Grades $\text{IT12}$ to $\text{IT16}$ are for coarse processes (casting, forging, stamping).

Mathematical Formulation of Tolerance Unit ($i$)

For basic sizes up to $500\text{ mm}$, the standard tolerance unit $i$ (in micrometers, $\mu\text{m}$) is given by:

$$\ni = 0.45\sqrt[3]{D} + 0.001 D$$

where:

• $D$ is the geometric mean diameter (in $\text{mm}$) of the diameter step boundaries $D_1$ and $D_2$: $$D = \sqrt{D_1 \cdot D_2}$$

Note: The term $0.45\sqrt[3]{D}$ accounts for machining errors, while $0.001 D$ accounts for measuring and thermal errors.

Values of Standard Tolerance Grades

For basic tolerance grades $\text{IT5}$ through $\text{IT16}$, the tolerance magnitude is a fixed multiple of $i$:

Notice that tolerance values increase in geometric progression with a common ratio $r = \sqrt[5]{10} \approx 1.585$, meaning the tolerance increases by a factor of 10 every 5 grades (e.g., $\text{IT11} = 10 \times \text{IT6}$). For sizes between $500\text{ mm}$ and $3150\text{ mm}$, the measuring unit is $I = 0.004 D + 2.1$.

---

3. Fundamental Deviations and Classification of Fits

There are 25 fundamental deviations represented by letters:

• Capital letters ($A, B, C, CD, D, E, EF, F, FG, G, H, J, JS, K, M, N, P, R, S, T, U, V, X, Y, Z$) for holes.
• Small letters ($a, b, c, cd, d, e, ef, f, fg, g, h, j, js, k, m, n, p, r, s, t, u, v, x, y, z$) for shafts.

Systems of Fits

1. Hole-Basis System ($H$-hole): The basic size is the lower limit of the hole. Lower deviation $EI = 0$. This is universally preferred in industrial practice because standardized fixed-size tools (drills, reamers, broaches) can be used to produce the hole, and shafts can be turned/ground to suit the desired fit.
2. Shaft-Basis System ($h$-shaft): The basic size is the upper limit of the shaft. Upper deviation $es = 0$. Used where a single continuous shaft diameter accommodates multiple components requiring different fits.

                                  FIT CLASSIFICATION
                                          |
        +---------------------------------+---------------------------------+
        |                                 |                                 |
  CLEARANCE FIT                    TRANSITION FIT                   INTERFERENCE FIT
  Hole > Shaft                      Hole ~ Shaft                      Shaft > Hole
  (e.g., H7/g6, H7/f7)             (e.g., H7/k6, H7/n6)              (e.g., H7/p6, H7/u6)

Quantitative Conditions for Fits

• Clearance Fit: The tolerance zone of the hole is entirely above that of the shaft. $$\text{Minimum Clearance} = LL_{\text{hole}} - UL_{\text{shaft}} = EI - es > 0$$ $$\text{Maximum Clearance} = UL_{\text{hole}} - LL_{\text{shaft}} = ES - ei$$
• Interference Fit: The tolerance zone of the shaft is entirely above that of the hole. $$\text{Minimum Interference} = LL_{\text{shaft}} - UL_{\text{hole}} = ei - ES > 0$$ $$\text{Maximum Interference} = UL_{\text{shaft}} - LL_{\text{hole}} = es - EI$$
• Transition Fit: The tolerance zones of the hole and shaft overlap. Depending on actual assembly sizes, either clearance or interference may result. $$\text{Maximum Clearance} = UL_{\text{hole}} - LL_{\text{shaft}} = ES - ei > 0$$ $$\text{Maximum Interference} = UL_{\text{shaft}} - LL_{\text{hole}} = es - EI > 0$$

---

4. Worked Calculation: Limits and Fits

Problem: A clearance fit is designated as $50\text{ mm } H7/g6$. The diameter step is $30\text{ mm}$ to $50\text{ mm}$. The fundamental deviation for shaft $g$ is $-2.5 D^{0.34}\;\mu\text{m}$. Calculate:

1. Fundamental tolerance unit $i$
2. Tolerances for hole and shaft
3. Limits of hole and shaft
4. Maximum and minimum clearances

Solution:

1. Mean diameter $D$:

   $$D = \sqrt{30 \times 50} = \sqrt{1500} \approx 38.7298\text{ mm}$$

2. Fundamental tolerance unit $i$:

   $$i = 0.45\sqrt[3]{38.7298} + 0.001(38.7298) = 0.45(3.3833) + 0.0387 = 1.5225 + 0.0387 = 1.5612\;\mu\text{m}$$

3. Hole Tolerance ($IT7$) and Limits:

   $$IT7 = 16i = 16 \times 1.5612 = 24.98\;\mu\text{m} \approx 0.025\text{ mm}$$

   For $H$-hole, fundamental deviation $EI = 0$. Therefore:

   $$LL_{\text{hole}} = 50.000\text{ mm}$$ $$UL_{\text{hole}} = 50.000 + 0.025 = 50.025\text{ mm}$$

4. Shaft Tolerance ($IT6$), Fundamental Deviation, and Limits:

   $$IT6 = 10i = 10 \times 1.5612 = 15.61\;\mu\text{m} \approx 0.016\text{ mm}$$

   Fundamental deviation for shaft $g$ ($es$):

   $$es = -2.5(38.7298)^{0.34} = -2.5(3.468) = -8.67\;\mu\text{m} \approx -0.009\text{ mm}$$ $$UL_{\text{shaft}} = 50.000 + es = 50.000 - 0.009 = 49.991\text{ mm}$$ $$LL_{\text{shaft}} = UL_{\text{shaft}} - IT6 = 49.991 - 0.016 = 49.975\text{ mm}$$

5. Clearance Calculations:

   $$\text{Minimum Clearance} = LL_{\text{hole}} - UL_{\text{shaft}} = 50.000 - 49.991 = +0.009\text{ mm} = 9\;\mu\text{m}$$ $$\text{Maximum Clearance} = UL_{\text{hole}} - LL_{\text{shaft}} = 50.025 - 49.975 = +0.050\text{ mm} = 50\;\mu\text{m}$$

---

5. Taylor's Principle of Limit Gauge Design

Fixed limit gauges (plug gauges for holes, snap/ring gauges for shafts) provide fast, Go/No-Go inspection on the assembly floor without reading scales.

                 TAYLOR'S GAUGE PRINCIPLE
                 ------------------------
  GO GAUGE                       NO-GO GAUGE
  - Checks Maximum Material      - Checks Least Material
    Condition (MMC)                Condition (LMC)
  - Checks Form & Assembly       - Checks Individual
    (Full contour/length)          Dimension (Point/Line)

Material Conditions

• Maximum Material Condition (MMC): The condition where a feature contains the maximum volume of material:
  • For a Hole: $\text{MMC} = \text{Minimum Hole Size} (LL_{\text{hole}})$
  • For a Shaft: $\text{MMC} = \text{Maximum Shaft Size} (UL_{\text{shaft}})$
• Least Material Condition (LMC): The condition where a feature contains the minimum volume of material:
  • For a Hole: $\text{LMC} = \text{Maximum Hole Size} (UL_{\text{hole}})$
  • For a Shaft: $\text{LMC} = \text{Minimum Shaft Size} (LL_{\text{shaft}})$

Taylor's Golden Rules

1. The GO Gauge: Must be designed to check the feature at its Maximum Material Condition (MMC) and check as many dimensions (full length, roundness, straightness) simultaneously as possible. It must enter the hole or pass over the shaft freely.
2. The NO-GO Gauge: Must be designed to check the feature at its Least Material Condition (LMC) and check only a single dimension (e.g., pin point contact or narrow blades) to prevent false acceptance due to out-of-roundness. It must not enter the hole or pass over the shaft.

Gauge Tolerances and Wear Allowance

• Gauge Maker's Tolerance: Allocated as $10\%$ of the work tolerance ($0.10 \times T_{\text{work}}$).
• Wear Allowance: Applied only to the GO gauge (as it rubs against every inspected component). A wear allowance of $5\%$ to $10\%$ of gauge tolerance is placed on the GO gauge within the work tolerance zone in the direction of wear.

---

6. Linear and Angular Metrology Instruments

Linear Metrology

• Vernier Calipers: Least Count ($LC$) is given by: $$LC = 1\text{ Main Scale Division (MSD)} - 1\text{ Vernier Scale Division (VSD)} = \frac{1\text{ MSD}}{n}$$ For standard metric calipers with $1\text{ MSD} = 1\text{ mm}$ and 50 divisions on the vernier scale: $LC = 1/50 = 0.02\text{ mm}$.
• Micrometer: Works on the principle of a precision nut and screw: $$LC = \frac{\text{Pitch of Spindle Screw}}{\text{Total Number of Thimble Divisions}} = \frac{0.5\text{ mm}}{50} = 0.01\text{ mm}$$
• Slip Gauges (Johannsen Gauges): Ultra-precision hardened steel or tungsten carbide blocks with mirror-finish parallel faces. Adhesion occurs via the wringing phenomenon (molecular attraction combined with a micro-thin oil layer). Calibration grades follow ISO 3650: Grade 00 (Reference master), Grade 0 (Calibration), Grade I (Inspection), Grade II (Workshop).

Angular Metrology: Sine Bar

A sine bar consists of a high-grade steel bar with two precision cylindrical rollers of equal diameter $d$ set at an exact center distance $L$ (typically $100\text{ mm}, 200\text{ mm},$ or $300\text{ mm}$). Setting a stack of slip gauges of height $h$ under one roller establishes an angle $\theta$:

$$\sin\theta = \frac{h}{L} \implies \theta = \arcsin\left(\frac{h}{L}\right)$$

               +-----------------------+ <-- Workpiece Top Face (Horizontal)
              /                       /
             /      SINE BAR         /
      (O)===+=======================+===(O)
       |                                 |
     Height                              |
      (h)                             Datum
       |                              Surface
   +-------+
   | Slip  |
   | Gauges|
   +-------+-----------------------------------

Sensitivity and Error Analysis of Sine Bar

Differentiating $\sin\theta = h/L$ with respect to $\theta$:

$$\cos\theta\,d\theta = \frac{dh}{L} = \frac{dh}{h/\sin\theta} \implies d\theta = \tan\theta\left(\frac{dh}{h}\right)$$

• For angles $\theta > 45^{\circ}$, $\tan\theta > 1$ and increases asymptotically toward infinity as $\theta \to 90^{\circ}$. A minute error in slip gauge height $dh$ produces a massive angular error $d\theta$.
• Hence, sine bars are strictly limited to angles $\le 45^{\circ}$ in precision metrology.

Autocollimator and Clinometer

• Autocollimator: An optical instrument that projects a parallel beam of collimated light onto a plane reflector and measures the angular tilt of the reflected beam via an eyepiece micrometer or CCD sensor. Used to check straightness, flatness of machine beds, and small angular deflections with resolution down to $0.1\text{ arc-second}$.
• Clinometer: An instrument utilizing a precision circular scale coupled to a spirit level or pendulum to measure absolute inclination angles of inclined surfaces over $0^{\circ}$ to $360^{\circ}$.

---

7. Surface Metrology and Roughness Parameters

Machined surfaces consist of two distinct deviations from the nominal geometry:

1. Waviness (Secondary Texture): Long-wavelength periodic irregularities caused by machine vibrations, chatter, or workpiece deflection.
2. Roughness (Primary Texture): Short-wavelength irregularities caused by the cutting tool feed marks, built-up edge fragmentation, and shear fracture.

  Peak ^             Roughness Profile y(x)
       |       /\          /\        /\
       |------/--\--------/--\------/--\---- Center Line (Mean Line)
       |     /    \  /\  /    \    /    \
       |____/______\/__\/______\__/______\__
  Valley                     Sampling Length (L)

Mathematical Definitions of Roughness Parameters

1. Arithmetical Mean Deviation ($R_a$ / CLA - Center Line Average): $$R_a = \frac{1}{L} \int_0^L |y(x)|\,dx \approx \frac{\sum_{i=1}^n |y_i|}{n}$$
2. Root Mean Square Roughness ($R_q$ / RMS): $$R_q = \sqrt{\frac{1}{L}\int_0^L y^2(x)\,dx} \approx \sqrt{\frac{\sum_{i=1}^n y_i^2}{n}}$$ For a pure sinusoidal roughness profile, $R_q = \frac{\pi}{2\sqrt{2}} R_a \approx 1.11 R_a$.
3. Ten-Point Height of Irregularities ($R_z$): The average difference between the 5 highest peaks and 5 deepest valleys within the sampling length: $$R_z = \frac{(y_{p1} + y_{p2} + y_{p3} + y_{p4} + y_{p5}) - (y_{v1} + y_{v2} + y_{v3} + y_{v4} + y_{v5})}{5}$$
4. Maximum Peak-to-Valley Height ($R_t$ or $h_{\max}$): The vertical distance between the highest peak crest and lowest valley trough.

Theoretical Surface Roughness in Machining

• For a Single-Point Turning Tool with Nose Radius $R_{\text{nose}}$ and Feed $f$: $$R_t = \frac{f^2}{8 R_{\text{nose}}}$$ The corresponding theoretical center-line average roughness is: $$R_a \approx \frac{R_t}{4} = \frac{f^2}{32 R_{\text{nose}}}$$ (where $f$ is in $\text{mm/rev}$ and $R_{\text{nose}}$ is in $\text{mm}$).
• For a Sharp Tool (Zero Nose Radius, Side Cutting Edge Angle $C_s$, End Cutting Edge Angle $C_e$): $$R_t = \frac{f}{\tan C_s + \cot C_e}$$ $$R_a = \frac{R_t}{4} = \frac{f}{4(\tan C_s + \cot C_e)}$$