Properties of Manufactured Products

(a) Unilateral and bilateral tolerance; geometric tolerance

Geometric tolerance limits the form, orientation, location and run-out of a feature, not just its size. It is given in a feature control frame on the drawing. Characteristic symbols:

• Straightness (a straight line) and flatness (a parallelogram) - form.
• Parallelism (two parallel lines) and perpendicularity (a right-angle) - orientation.
• Concentricity (two circles) and position - location.
• Circular run-out (arrow) - run-out.

(b) Fit calculation

Hole: maximum size $= 40.039$ mm, minimum size $= 40.000$ mm.

Maximum clearance $=$ hole max $-$ shaft min; minimum clearance $=$ hole min $-$ shaft max (negative value = interference).

Shaft (i): shaft max $= 39.975$, min $= 39.950$

$$C_{max} = 40.039 - 39.950 = 0.089\ \text{mm},\qquad C_{min} = 40.000 - 39.975 = 0.025\ \text{mm}$$

Both positive, so it is a clearance fit (running fit).

Shaft (ii): shaft max $= 40.059$, min $= 40.043$

$$C_{max} = 40.039 - 40.043 = -0.004\ \text{mm},\qquad C_{min} = 40.000 - 40.059 = -0.059\ \text{mm}$$

Both negative, so maximum interference $= 0.059$ mm and minimum interference $= 0.004$ mm: an interference (press) fit.

Shaft (iii): shaft max $= 40.042$, min $= 40.026$

$$C_{max} = 40.039 - 40.026 = 0.013\ \text{mm},\qquad C_{min} = 40.000 - 40.042 = -0.042\ \text{mm}$$

The result can be clearance or interference, so it is a transition fit.

Answer: (i) clearance fit, 0.089 to 0.025 mm; (ii) interference fit, 0.004 to 0.059 mm; (iii) transition fit, 0.013 mm clearance to 0.042 mm interference.

Surface texture has roughness (fine irregularities from the process), waviness (longer wavelength, from vibration or deflection) and lay (direction of the pattern).

Definitions

$R_a$ is the arithmetic mean of the absolute deviations of the profile from the mean line:

$$R_a = \frac{1}{n}\sum_{i=1}^{n}|y_i|$$

$R_q$ (also RMS) is the square root of the mean of the squares of the deviations:

$$R_q = \sqrt{\frac{1}{n}\sum_{i=1}^{n} y_i^2}$$

Check of the mean line

Sum of ordinates: $2.0-1.5+3.5-2.5+1.0-3.0+2.5-1.0+3.0-4.0 = 0$, so the reference line is the mean line.

Calculation of $R_a$

$$\sum|y_i| = 2.0+1.5+3.5+2.5+1.0+3.0+2.5+1.0+3.0+4.0 = 24.0$$ $$R_a = \frac{24.0}{10} = 2.4\ \mu\text{m}$$

Calculation of $R_q$

$$\sum y_i^2 = 4+2.25+12.25+6.25+1+9+6.25+1+9+16 = 67.0$$ $$R_q = \sqrt{\frac{67.0}{10}} = \sqrt{6.7} = 2.59\ \mu\text{m}$$

Maximum peak-to-valley height

$$R_{max} = y_{max} - y_{min} = 3.5 - (-4.0) = 7.5\ \mu\text{m}$$

Factors causing roughness in machining

1. Feed marks: in turning the theoretical roughness is $R_{max} = f^2/(8r)$ where $f$ is the feed and $r$ the nose radius.
2. Built-up edge, tool wear, vibration (chatter) and machine tool rigidity.

Answer: $R_a = 2.4\ \mu$m, $R_q = 2.59\ \mu$m, $R_{max} = 7.5\ \mu$m.

Residual stresses are stresses that remain in a body after all external loads and temperature gradients have been removed. They are self-balancing: tensile stress in one region is balanced by compressive stress in another.

Origin

Residual stress appears whenever different parts of a body deform plastically by different amounts, or cool at different rates.

1. Non-uniform plastic deformation: in bending, the outer fibres are plastically stretched and when the load is removed the elastic core pushes them back, leaving compressive stress on the surface that was in tension. Rolling, drawing and shot peening also leave surface compression.
2. Thermal gradients: in quenching, the surface cools first and contracts; later the core contracts against a rigid shell, leaving surface compression and core tension. In welding, the weld region shrinks against cold parent metal and is left in tension.
3. Phase transformation: martensite formation causes volume expansion in some regions (hardening of steel).
4. Machining and grinding: mechanical deformation under the tool and heat generated leave a thin surface layer in tension (grinding heat) or compression (sharp tools, low feed).

Effects

• Distortion and warping when material is machined away or heated.
• Compressive residual stress at the surface improves fatigue life and resists crack growth.
• Tensile residual stress at the surface reduces fatigue strength and causes stress-corrosion cracking and cold cracks in welds.
• Dimensional instability of precision parts with time.

Methods of reduction or removal