Machining and Machine Tools

Metal cutting mechanics

In orthogonal machining, the cutting edge is perpendicular to the cutting velocity, and the chip flows over the rake face. The chip thickness ratio, or cutting ratio, is r = t_1/t_2, where t_1 is the undeformed chip thickness and t_2 is the deformed chip thickness. Its reciprocal r_c = t_2/t_1 is the chip compression ratio, which is always greater than one. Using the Merchant relation, the shear angle follows from \tan\phi = \dfrac{r\cos\alpha}{1 - r\sin\alpha}, where \alpha is the rake angle. For a zero-rake tool, this reduces to \tan\phi = r. A larger rake angle therefore gives a larger shear angle for the same cutting ratio.

The forces on the rake face are the cutting force F_c and the thrust force F_t. The friction force is F = F_t\cos\alpha + F_c\sin\alpha and the normal force is N = F_c\cos\alpha - F_t\sin\alpha, so the friction coefficient is \mu = F/N. At zero rake angle, F = F_t and N = F_c, so \mu = F_t/F_c. For a general rake angle, \mu = \dfrac{F_t + F_c\tan\alpha}{F_c - F_t\tan\alpha}.

The cutting power is P_c = F_c\,v, and the specific cutting energy u is the power per unit volume removal rate, so P_c = u\,\text{MRR}. In turning, \text{MRR} = v\,f\,d with v in mm/s, f in mm/rev and d in mm. The machining time for turning is the total tool travel including approach and overrun, divided by the feed rate fN, where the spindle speed follows from v = \pi DN/1000.

In shaping, each stroke consists of the cutting stroke and the return stroke, and the return time is the cutting time multiplied by the return-to-cutting ratio. The number of strokes is the width of the surface divided by the feed per stroke. The width of cut is b = d/\sin\kappa, where d is the depth of cut and \kappa is the principal cutting edge angle. In orthogonal cutting, \kappa = 90^\circ, so the width equals the depth, and the feed changes only the undeformed chip thickness.

Merchant's minimum-energy condition gives \phi = C - \dfrac{\alpha}{2} + \dfrac{\beta}{2}, where \beta is the friction angle. If the shear stress on the shear plane does not depend on the normal stress, C takes the universal value \pi/4. If the shear stress depends linearly on the normal stress, C becomes a function of material constants.

Taylor tool life equation

Taylor's tool life equation VT^n = C relates the cutting speed V to the tool life T, where n is the tool life exponent and C is a constant for a given tool-work combination. The exponent n is small for high-speed steel and becomes larger for carbide and ceramic tools, which tolerate higher cutting speeds. For two cutting conditions, V_1T_1^n = V_2T_2^n, so the life ratio is \dfrac{T_2}{T_1} = \left(\dfrac{V_1}{V_2}\right)^{1/n}. If the cutting speed is doubled and the life falls to one-eighth of its original value, the exponent is n = 1/3.

In turning with a fixed workpiece diameter, the cutting speed is proportional to the spindle speed, so two speed–life pairs can be used to find n directly from the ratio of lives. Once n is known, the life at any other speed follows from the same proportionality.

The modified Taylor equation includes the feed, VT^nf^m = \text{constant}. When two conditions give the same tool life, T_1 = T_2, so the life terms cancel and V_1f_1^m = V_2f_2^m. Hence \left(\dfrac{f_1}{f_2}\right)^m = \dfrac{V_2}{V_1}, and the feed ratio is found by raising the speed ratio to the power 1/m.

Nontraditional machining processes

Nontraditional machining removes material by mechanisms other than conventional shear, and each process is suited to particular features and materials. In chemical machining, the etchant attacks the workpiece in all directions, so the material under the edge of the mask is also removed. This undercutting reduces dimensional accuracy, and chemical machining is therefore not suited to features that need tight tolerances.

Water jet machining cuts soft materials, including edible items, with a high-velocity jet of water and produces no heat-affected zone. Electric discharge machining is well suited to deep square blind holes in conductive materials such as mild steel, because the eroded material is carried away by the dielectric. Laser beam machining is used for high-speed profile cutting of thin sheets.

In laser beam machining, the predominant removal mechanism is melting and vaporization of a small localized zone. Abrasive water jet machining removes material by impact-based erosion from high-velocity abrasive particles. Ion beam machining removes atoms by sputtering, which is a momentum transfer process. Mechanical milling removes material by shearing, as in conventional milling.

Rotary-to-linear motion mechanisms

Machine tools often need to convert rotation into translation. A screw and nut pair converts the rotation of the screw into linear travel of the nut and is widely used in feed drives. A cam and follower converts the continuous rotation of the cam into a defined translational or reciprocating motion of the follower. The Whitworth quick-return mechanism converts the rotary motion of a crank into the reciprocating motion of a ram, as used in shaping machines.

A four-bevel-gear differential does not produce translation. It combines or divides rotational speeds between shafts and transmits rotation to rotation, so it is not used for rotary-to-linear conversion.

EDM discharge energy

In electric discharge machining, the gap between the tool and the workpiece behaves as a capacitor that charges to the discharge voltage V_b before breakdown. The energy released in each spark is therefore E = \tfrac{1}{2}CV_b^2, where C is the circuit capacitance. The energy per spark is proportional to the square of the discharge voltage, so doubling the voltage quadruples the energy of each spark, while the capacitance is held constant.

Surface roughness parameters

Surface roughness in turning depends on the feed, the tool geometry and the nose radius. For a sharp tool with zero nose radius, the ideal maximum height of unevenness is R_{max} = f/(\tan\psi + \cot\psi'), where \psi and \psi' are the side and end cutting edge angles, and f is the feed per revolution. The height is proportional to the feed and falls as the cutting edge angles are increased.

When the tool has a nose radius r, the ideal peak-to-valley height is R_{max} = f^2/(8r). The height therefore varies with the square of the feed and falls in inverse proportion to the nose radius. A tool with a nose radius gives a much smaller ideal height than a sharp tool at the same feed.

CNC programming and drives

An open loop positioning system uses a stepper motor, which moves a fixed angle for each pulse it receives, and it needs no position feedback. The number of pulses per revolution is 360^\circ divided by the step angle. If a gear train connects the motor to a lead screw, the speed ratio between them determines how many motor revolutions produce one lead-screw revolution. The table travel per pulse equals the screw pitch multiplied by the number of lead-screw revolutions per pulse. The number of pulses for a required travel is the travel divided by the travel per pulse.