CNC Axis Drive Sizing: Ballscrew, Belt, Stepper and Servo

For peripheral milling, feed per tooth is $f_z=v_f/(nz)$. Specific cutting force $k_c$ is energy per removed volume in equivalent force/area units. A flute at engagement angle $\phi$ removes chip thickness $h=f_z\sin\phi$; $dF_t=k_c h\,dz$, $dF_r=k_r dF_t$ and $dF_z=k_a dF_t$. The tool derives the force projections and analytically integrates them along each uniform helix, with phase lag $2z\tan\beta/D$. Climb engagement is $[\pi-\gamma,\pi]$, conventional engagement $[0,\gamma]$, where $\gamma=\cos^{-1}(1-2a_e/D)$. The planar resultant bounds either horizontal-axis projection. The cutting allowance multiplies the absolute selected-axis waveform. One tooth-passage period is partitioned at flute entry, exit, and their helix-shifted events. Local refinement samples both sides of every boundary and the interval interior for peak force; weighted four-point Gauss quadrature in subintervals computes phase averages without missing narrow engagements. Sandvik's milling formulas (attached p. 1) independently give mean power from $k_c a_e a_p v_f$.

Guide force is $\mu M g\cos\theta+F_{drag}$, while signed gravity and counterbalance contribute $Mg\sin\theta+F_{bias}$. Add $Ma$ once. The transmission radius is $r=L/(2\pi)$ for a ballscrew and half the pulley or pinion pitch diameter otherwise. Motoring torque is $Fr/\eta$ plus preload; an overhauling load contributes $Fr\eta_{back}$. Screw/pulley rotational inertia is added at the output shaft, reduction losses follow power direction, then rotor and input-side inertia are accelerated at the motor. THK selection examples, attached pp. 4–6 and 12–13, and Oriental Motor's sizing examples support this separation.

Physical reflected load inertia is $J_L=(Mr^2+J_{out})/R^2+J_{input}$, excluding rotor inertia and efficiency. Both candidates are checked at every sampled cutting and rapid speed, including both travel directions and acceleration signs. Cutting force scales with feed along the speed sweep. The separate torque service factor is applied to motor demand.

The electrical stepper estimate maximizes q-axis current inside both the sine-current circle and the steady-state voltage circle: $v_d=Ri_d-\omega_e L i_q$, $v_q=Ri_q+\omega_e L i_d+k_e\omega_m$. In SI, $k_e=k_t=T_{sine}/I$. It includes possible negative d-axis current at the voltage limit. The usable bus fraction is 0.9; a separate utilization input derates this ideal synchronous envelope. Oriental Motor Stepper Technical Manual, chapter 7, pp. 83–89, gives the underlying voltage and torque equations. Entered pull-out points instead use PCHIP interpolation over their measured speed range.

For the servo, continuous and peak torque are constant to base speed and constant-power above it. RMS torque integrates torque squared over each cutting and rapid trapezoidal/triangular phase and dwell. Cutting resistance follows instantaneous feed, including short-stroke peak feed. Each ramp is split where linear-load or output-shaft power changes sign; torque is affine in feed on each segment, so its squared time integral is evaluated exactly. The overload check finds the longest contiguous moving/dwell block whose conservative peak exceeds continuous torque.

For a steel ballscrew, first-mode bending gives $\omega_{cr}=\beta_1^2\sqrt{EI/(\rho A)}/L_b^2$. The speed allowance is 0.8 of critical, further limited by DN and the rated drive speed. Euler compression capacity uses THK mounting multipliers 0.25, 1, 2 or 4 and a 0.5 factor; also check the 147 MPa permissible axial stress and entered allowable thrust. Whirling and compression spans/fixity are separate. THK and HIWIN selection guides use these checks.

Uniform, equally spaced, rigid helical flutes and constant effective kc; 90° peripheral side cutting with axial depth no greater than ten cutter diameters. The material presets are Sumitomo's rough planning coefficients, not grade-specific cutting data. Radial/tangential ratio 0.3, axial/tangential ratio 0.2, cutting allowance 1.25, and electrical utilization 0.65 are editable design assumptions.

The screw is a solid steel cylinder for rotating inertia and a root-diameter cylinder for bending and compression: E = 206 GPa, density = 7850 kg/m³. Nut mass belongs in moving mass. Additional input-side coupling/reducer inertia is editable in the stepper specification table and applies to both candidates. Guide normal force includes weight normal to the guide only; add measured drag when cutting reactions or rail preload increase resistance.

The default stepper electrical basis is the 23HS30-2804S revision-1 data; sine-current torque is two-phase holding torque divided by √2 at the same phase-current peak. The default resistance is the maker's 25 °C value: enter the operating-temperature resistance for a hot motor. The inertia-ratio limit of 10 is a planning assumption, not a universal stepper rating. The default servo torque ratings and rotor inertia come from Oriental Motor's NXM620A example; its permitted maximum speed is conservatively capped at the cited 3000 rpm rated speed. The 1 s peak-duration setting, command counts, drive-thrust rating and controller pulse limit are design inputs that must match the actual equipment.

This is axis drive sizing for peripheral milling, not a toolpath, chatter or structural-dynamics simulation. It excludes edge/ploughing forces, runout, variable helix, tool bending, chip recutting, jerk, static breakaway friction, changing cable loads, ballnut life, bearing life, alignment, and belt/rack tooth or pretension design. Resolve both screw spans at every nut position and use the governing case; a nut is not automatically a lateral support. Euler buckling applies to slender steel segments; short shafts need a direct strength check. Ballnuts, supports and recirculation systems can impose lower catalog limits.

The cutting-axis sign conventions belong to the idealized cutter geometry. Absolute force is treated as resisting travel, and the planar resultant is conservative for either horizontal axis. Vertical-axis gravity, counterbalance and motor holding torque are included; brake holding, loss of power, regenerative bus-voltage rise and stopping protection require separate design.

The electrical stepper curve is an estimate, not a measured pull-out curve or a guaranteed motor recommendation. It omits resonance, instability, saturation, iron losses and driver bandwidth. Use the manufacturer's pull-out curve for the exact driver, current, wiring and voltage before selecting hardware. No curve is invented for the entered-data mode: its zero placeholders must be replaced. The servo envelope must likewise match the real motor/drive; its duration screen does not replace a drive I²t/thermal model.

The envelope checks commanded speeds even when a short stroke cannot reach them. Duty RMS scales cutting resistance with instantaneous feed during both ramps and uses the achieved peak feed during cruise. Full nominal ramp resistance is not a conservative assumption on an overhauling axis because cutting can reduce braking torque. The continuous-capacity comparison conservatively uses the lowest servo continuous torque over the commanded speed range. Cutting-force ripple is averaged over spindle phase for heating; peak torque is checked separately.

The default phase target is 720 intervals per full revolution. Identical flutes make the force periodic over one tooth pitch, which is shown in the chart. Engagement events always split the integration domain, with at least 16 Gauss nodes and 16 peak-sampling intervals in every nonzero event interval, however narrow. An additional doubled peak grid and a comparison against kc·MRR trigger a warning if peak refinement exceeds 0.1% or cutting-power error exceeds 0.001%. Increase the advanced resolution until the warning clears. The torque-speed sweep uses at least 401 points and follows the phase target. Axial helix integration and the piecewise-quadratic time integral during feed ramps are analytic; neither uses a depth or time-step mesh. Nominal microstep/encoder increments are command resolution, not positioning accuracy. A reserve above one means only that the modeled check is met.