Machining, Metrology and CNC

Machining, CNC and surface finish

Surface generation begins with the motion of the cutting edge. A surface is generated by a generatrix, the profile traced by the tool, moving along a directrix. When a plane curve is translated along a straight line normal to its own plane, every section parallel to the profile is identical, so the surface is a cylinder. A straight generatrix moved along a line lying in its own plane gives a plane, whereas a curve revolved about an axis gives a surface of revolution. Always decide whether the motion is translation or rotation before naming the surface. Hole machining follows a fixed order of progressively finer operations. Drilling makes the initial hole in solid stock, boring enlarges an existing hole and improves its size, alignment and roundness, and reaming is the final step that removes a small allowance to give close size and fine finish. A pre-existing hole is needed before boring or reaming, so drilling always comes first. Backlash is the clearance between mating parts of a feed drive, and it causes a position error whenever the direction of motion reverses. Preloaded ballscrews use an axial preload, often through a double nut, to take up the clearance between balls and raceways, which is why modern CNC axes are built this way. Rack-and-pinion, ratchet and slider-crank arrangements do not provide such a preload. Centre-line average roughness is R_a = \frac{1}{L}\int_0^L |y(x)|\,dx, the mean of the absolute deviations from the mean line over the sampling length. A profile whose height is constant in magnitude gives R_a equal to that height, regardless of how peaks and valleys alternate in sign, so R_a is an averaged measure and not a peak-to-valley measure. Specific grinding energy u is grinding power divided by material removal rate. Power equals u \times \text{MRR}, and the tangential force follows from F_t = P/V_s with V_s = \pi D N. Only the tangential force does work, so the normal force does not enter u. The contact width is limited by the smaller of the wheel width and the workpiece width, so for a wheel wider than the part, the workpiece width governs the removal rate. When a backlash is stated as an angle on a lead screw, the linear backlash per revolution is pitch \times angle/360. The conventional treatment multiplies this per-revolution value by the number of screw revolutions covered by the travel. An interpolator inside the CNC controller computes intermediate coordinates between programmed end points along lines or arcs, producing a sequence of reference positions that prescribe the shape the tool must follow. It is distinct from tool radius compensation, machine auxiliary control such as lubrication, and CAM post processing. For a stepper-driven lead screw, the pulse rate equals (table speed \div lead) \times pulses per revolution. Converting the table speed to mm/s before dividing by the lead gives revolutions per second, and multiplying by the pulses per revolution gives the pulse frequency, assuming no missed pulses. In the standard marking of a grinding wheel, the grit number follows the abrasive type, and a larger grit number means a finer grain. With the other designation letters unchanged, the finest grit gives the smallest scratches and the best surface finish, while a coarse grit removes material faster. In CNC part programs, address letters identify word types. G denotes preparatory functions that set the motion or mode, such as G01 for linear interpolation, G02 and G03 for circular interpolation, and G90 and G91 for absolute and incremental positioning. M denotes miscellaneous functions such as spindle and coolant control, O identifies the program number, and P carries parameters such as dwell time. The ten-point height R_z is the average of the five highest peak heights and the five deepest valley depths, measured from a reference line parallel to the general lay. Because it depends on the extreme peaks and valleys, it is more sensitive to large deviations than R_a, so the two groups must be taken separately and each averaged over five.

Orthogonal machining force analysis

In orthogonal cutting the cutting edge is perpendicular to the cutting velocity, so the chip flows in a single plane. The chip thickness ratio is r = t_0/t_c = V_{chip}/V_c, where t_0 is the uncut chip thickness and t_c is the chip thickness. For a zero rake angle, \tan\phi = r, so the shear angle follows directly from the measured chip velocity. The shear plane area is A_s = t_0 w/\sin\phi, where w is the width of cut, and the shear strength is \tau = F_s\sin\phi/(t_0 w). In general, \tan\phi = \dfrac{r\cos\alpha}{1 - r\sin\alpha}, where \alpha is the rake angle. The cutting force F_c acts along the velocity and the thrust force F_t acts normal to it. The shear force is F_s = F_c\cos\phi - F_t\sin\phi. Merchant's relation F_c = F_s\dfrac{\cos(\beta-\alpha)}{\cos(\phi+\beta-\alpha)} gives the cutting force directly from the shear force when the friction angle \beta and the shear angle \phi are known. The friction force along the rake face is F = F_c\sin\alpha + F_t\cos\alpha, so the thrust force can be found as F_t = (F - F_c\sin\alpha)/\cos\alpha. Substituting this into F_s = F_c\cos\phi - F_t\sin\phi gives the shear force, which is the route used when the friction force and the cutting force are the known quantities. Cutting power equals P = F_c V. In orthogonal turning of a thin tube whose wall thickness is the width of cut, the removal rate equals \pi D F_r t, where F_r is the axial feed rate per minute and t is the wall thickness. This rate is independent of the rotational speed, and power is found as P = u \times \text{MRR} with specific energy u. When the cutting force and thrust force are equal, F_s = F_c(\cos\phi - \sin\phi) for a zero rake tool, so F_c = \dfrac{\tau t_0 w}{\sin\phi(\cos\phi - \sin\phi)}. This simplification of the shear-force relation is a common step in orthogonal problems. In the ideal orthogonal model the shear zone is a thin layer of thickness \delta. The velocity of material across the shear plane is V_s = \dfrac{V\cos\alpha}{\cos(\phi-\alpha)}, so the shear strain rate is \dot\gamma = V_s/\delta = \dfrac{V\cos\alpha}{\delta\cos(\phi-\alpha)}. The shear strain itself is \gamma = \cot\phi + \tan(\phi-\alpha).

Limits, fits and tolerances

The tolerance of a dimension is the difference between its upper and lower limits. In a hole and shaft pair, the type of fit depends on how the limits compare. A clearance fit occurs when the smallest hole is larger than the largest shaft, and an interference fit occurs when the largest hole is smaller than the smallest shaft. When the two tolerance zones overlap, the fit is a transition fit, in which either a small clearance or a small interference can result. Allowance is the intentional minimum clearance between hole and shaft, and it equals the lower limit of the hole minus the upper limit of the shaft. The maximum clearance equals the upper limit of the hole minus the lower limit of the shaft. The maximum material limit of a hole is its lower limit, the smallest hole size, while the maximum material limit of a shaft is its upper limit. A negative minimum clearance signals that interference is possible, and a positive maximum clearance signals that clearance is possible. When both occur, the fit is classified as transition.

Taylor tool life equation

Taylor's tool life equation is VT^n = C, where V is the cutting speed, T is the tool life, n is an exponent that depends on the tool and work material, and C is a constant. Equivalently, T = (C/V)^{1/n}. Taylor's equation lets tool life at one speed be found from a known life at another speed by T_2 = T_1(V_1/V_2)^{1/n}. From two tests at different speeds, the exponent follows from n = \dfrac{\ln(V_2/V_1)}{\ln(T_1/T_2)}. Once n is known, the tool life at any speed is obtained from the same relation, and the exponent is larger for tool materials that tolerate higher temperatures. When tool life is counted as a number of holes, the life in time equals the number of holes multiplied by the time per hole. With feed per revolution and depth fixed, the time per hole is inversely proportional to the rotational speed, so the ratio of lives gives the exponent through the same logarithmic formula. For turning, the machining time per piece is T_m = \dfrac{\pi D L}{fV}, with V and f in consistent units. The number of tool changes per piece is T_m/T, so the tool changing cost per piece is \dfrac{T_m}{T}\,t_c\,M, where t_c is the tool change time and M is the labour and overhead rate per minute.

Non-traditional machining processes

Ablation is the removal of material by melting and vaporization produced by a concentrated energy source. Laser beam machining is the standard example: a focused high-power beam heats a small zone until the material melts and vaporizes. The other non-traditional processes remove material by different mechanisms, such as erosion from an abrasive jet, chemical etching in a reagent, anodic dissolution in electrochemical machining, and spark erosion in electric discharge machining. Ceramics and glass are hard and brittle, and they are often electrically non-conductive, so processes that depend on conductivity or cause thermal cracking are unsuitable. Ultrasonic machining is the conventional choice: a tool vibrates at high frequency in an abrasive slurry and removes material by repeated impacts that fracture the brittle surface. It works regardless of electrical conductivity.