Machining

Machining Types

Milling: Rotating tool removes material from stationary workpiece.
Turning: Stationary tool removes material from rotating workpiece.
Drilling: Rotating drill bit creates holes in the workpiece.

Cutting Geometry

image of cutting geometry (slide 16)

Terms: Rake angle $ \alpha $, Shear angle $ \phi $, Chip thickness $ t_c $, Depth of cut $ t_o $, Length of shear plane $ l_s $
Chip ratio: $ r = \frac{t_o}{t_c} $
Shear angle: $ \tan \phi = \frac{r \cos \alpha}{1 - r \sin \alpha} $

Cutting Forces

image of cutting forces (slide 18/19)

Friction angle $ \beta $, Coefficient of friction $ \mu = \tan \beta $, Resultant $ R $

Tool-Chip Interface

Friction force $ F = R \sin \beta $
Normal Force $ N = R \cos \beta $

Vertical & Horizontal

Thrust force $ F_t = R \sin(\beta - \alpha) $
Cutting force $ F_c = R \cos(\beta - \alpha) $

Shear Plane

Normal force $ N_s = R \sin(\beta - \alpha + \phi) $
Shear force $ F_s = R \cos(\beta - \alpha + \phi) $
Shear Stress $ T = \frac{F_s}{A_s} = \frac{R \cos(\beta - \alpha + \phi) \sin\phi}{w t_0} $

*box

To minimize the shear stress, differentiate with respect to the shear plane angle $ \phi $ and set the derivative equal to zero:

$$ \frac{d}{d\phi}\left(\cos(\beta - \alpha + \phi)\sin\phi\right) $$

Differentiation with Respect to phi

$$ 0 = \frac{d}{d\phi}\left(\frac{R}{w t_o}\cos(\beta - \alpha + \phi)\sin\phi\right) $$

$$ \frac{d}{d\phi}\left(\cos(\beta - \alpha + \phi)\sin\phi\right) $$

Applying the product rule:

$$ 0 = \cos(\beta - \alpha + \phi)\cos\phi - \sin(\beta - \alpha + \phi)\sin\phi $$

Rearranging terms:

$$ \sin(\beta - \alpha + \phi)\sin\phi = \cos(\beta - \alpha + \phi)\cos\phi $$

Dividing both sides:

$$ \tan(\beta - \alpha + \phi) = \frac{\cos\phi}{\sin\phi} = \cot\phi $$

Using identity:

$$ \tan(\beta - \alpha + \phi) = \tan(90^\circ - \phi) $$

Equating arguments:

$$ \beta - \alpha + \phi = 90^\circ - \phi $$

Solving for phi:

$$ \boxed{\phi = 45^\circ - \frac{\beta}{2} + \frac{\alpha}{2}} $$

This proves: Increase in rake angle ($ \alpha $) and decrease in friction angle ($ \beta $)

higher shear angle
lower shear plane area
less force needed
lower power and temperature

$$ P = u_t MRR = 1.64 \text{ hp} $$

$$ F_c = \frac{P}{V} = 349 \text{ lbf} $$

$$ t = \frac{l}{fN} = 0.25 \text{ min} $$

Example Problem (Metric)

$ l = 150 $ mm
$ D_i = 10 $ mm, $ D_f = 8 $ mm
$ N = 400 $ rpm
Feed speed = $ 200 $ mm/min
$ u_t = 2.8 $ N$ \cdot $m/mm$ ^3 $

$$ f = 0.5 \text{ mm/rev}, \quad D_{avg} = 9 \text{ mm} $$

$$ MRR = 5650 \text{ mm}^3/\text{min} $$

$$ t = 0.75 \text{ min} $$

$$ P = 264 \text{ W}, \quad T = 6.3 \text{ N}\cdot\text{m} $$

Drilling

Terms: bit diameter $ D $, feed (len/rev) $ f $, drill speed (rev/time) $ N $

$ MRR = \frac{\pi D^2}{4} f N $

Recommended drilling speeds and feed rates (insert chart from slide 70)

Example Problem

$ D = 15 $ mm
$ f = 0.1 $ mm/rev
$ N = 500 $ rpm

$$ MRR = 8840 \text{ mm}^3/\text{min} $$

$$ P = 58.9 \text{ W}, \quad T = 1.13 \text{ N}\cdot\text{m} $$

Tool Life

Terms: cutting speed $ V $, tool life $ t $, constants $ n $ and $ C $

Taylor equation: $ Vt^n = C $

Example Problem

$$ Vt^{0.2} = C $$

If $ V $ is reduced by 30%

$$ \frac{t_2}{t_1} = \left(\frac{1}{0.7}\right)^5 \approx 6 $$

Tool life increases by approximately 6 times.

Surface Roughness

Depends on feed rate, tool geometry, and material
Finishing operations improve surface quality

Surface Tolerances (insert chart from slide 80)

Machining Economics

*Callout: Maximizing production rate minimizes cutting time per unit.

Terms: total time per unit product for operation $ t_c $, part handling time per part $ t_h $, machining time per part $ t_m $, tool change time $ t_t $, number of pieces cut in one tool life $ n_p $

Total cycle time: $ t_c = t_h + t_m + \frac{t_t}{n_p} $

Terms: cost rate for operator and machine $ C_o $, cost of part handling time $ C_o t_h $, cost of tool change time $ \frac{C_o t_t}{n_p} $, cost per cutting edge $ C_t $, tooling cost $ \frac{C_t}{n_p} $

Total cost per unit product for operation: $ C_c = C_o t_h + C_o t_m + \frac{C_o t_t}{n_p} + \frac{C_t}{n_p} $

Design Advisor

Recommended vs. Avoided Design Aspects.