Motion hardware

Belts — three modes.

Screws — trapezoidal (Tr, 30°), ACME (29°), square and ball screws. Torque to drive a load, efficiency, whether it is self-locking or back-drives (and the holding torque if it does), the torque to lower the load, linear speed, critical (whirling) speed and buckling load for the end fixity, and a chart of efficiency against lead angle for your friction. Friction defaults: 0.1 for Tr with a bronze or plastic nut, 0.15 for ACME and square, 0.003 for ball screws. Leave the root diameter at 0 to estimate it (ISO 2904 for Tr; D − P for ACME and square; 0.8 d for ball screws — use the catalogue figure). For steps per mm from the lead, see the stepper tool.

DC motor — from the datasheet’s voltage, no-load speed and current, stall torque and current: Kt, Kv, winding resistance, peak power and peak efficiency, speed–current and power–efficiency charts against torque, and which gearbox ratios can deliver a load torque at a load speed (with the one drawing least current marked). A warning appears if the figures do not agree with each other (Kt ω₀ should equal V − I₀R). A small Kv → Kt converter sits underneath.

Servo — a hobby servo’s rated torque (kg·cm, as sold) and arm length → force at the tip, the heaviest load it can hold there, the torque your load needs at an angle, and how much of the rating that uses with a safety factor. Also in N·m and oz·in.

Will it move? — moving mass, acceleration, top speed, friction, incline and any extra force → the force on the carriage (broken down), the motor torque through a belt pulley or a screw including the rotor’s own inertia, the motor speed, and a verdict against the motor torque you have at that speed with a safety factor.

Belt length and centre distance: the open-belt equations in Shigley’s Mechanical Engineering Design (§17-2) and the Gates PowerGrip GT/HTD design manuals; at least six teeth in mesh is Gates’s guideline.

Power screws: Shigley §8-2. With tan ρ′ = f ÷ cos α and λ = atan(lead ÷ π d_m): raise T = F d_m/2 · tan(λ + ρ′), lower T = F d_m/2 · tan(ρ′ − λ), efficiency tan λ ÷ tan(λ + ρ′); collar T_c = F f_c d_c/2. It reproduces Shigley’s Example 8-1 (32 mm square thread, double start, f = 0.08, 6.4 kN): 15.94 N·m for the thread, 26.18 N·m with the collar, −0.47 N·m to lower, overall efficiency 0.31. A screw is self-locking when ρ′ ≥ λ.

Ball screws: THK’s forward efficiency η₁ = (1 − μ tan β)/(1 + μ/tan β) and reverse η₂ = (1 − μ/tan β)/(1 + μ tan β) — algebraically the same as the power-screw formula with α = 0.

Critical speed n = λ₂ d_r/L² × 10⁷ rpm and buckling P = η₂ d_r⁴/L² × 10⁴ N (THK and Hiwin ball-screw catalogues; λ₂ = 3.4, 9.7, 15.1, 21.9 and η₂ = 1.3, 5, 10, 20 for fixed–free, supported–supported, fixed–supported and fixed–fixed, with 0.8 and 0.5 safety factors built in). They agree with Euler–Bernoulli whirling and Euler buckling for a steel shaft to about 1% and 4%.

DC motors: the linear model in maxon’s "Key information" pages — speed falls linearly from n₀ at no load to zero at stall; I = I₀ + T ÷ Kt with Kt = T_stall ÷ (I_stall − I₀); Kt [N·m/A] = 60 ÷ (2π Kv [rpm/V]) ≈ 9.549 ÷ Kv; peak power T_stall ω₀ ÷ 4 at half stall torque; peak efficiency (1 − √(I₀/I_stall))² at I = √(I₀ I_stall). A gear ratio i works when n_load i ≤ n₀ (1 − T_load ÷ (i η T_stall)).

Servos: torque at the tip = rating ÷ arm; a mass m at angle θ needs m g r cos θ. 1 kg·cm = 0.0980665 N·m = 13.887 oz·in.

Linear axis: F = m a + m g (sin θ + μ cos θ) + F_extra; belt T = F r ÷ η with r = teeth × pitch ÷ 2π; screw T = F × lead ÷ (2π η); rotor J α with α = a ÷ r or 2π a ÷ lead.

Belt OD uses catalogue pitch-line differentials; tooth profiles (GT2 vs GT3, T5 vs AT5) differ in detail, so buy pulleys and belts of the same profile. The GT2 loop list is what is commonly sold, not a standard — check your supplier.

Screw friction is the biggest uncertainty: a dry plastic nut can be twice the default, a lubricated bronze nut less. Back-driving also depends on vibration — a "self-locking" screw near the limit can creep. Collar friction is zero when the screw runs in a rolling thrust bearing.

Critical-speed and buckling figures use the root diameter of a solid steel shaft (E = 206 GPa) and the catalogue safety factors; the nut’s own speed limit (DN value) and the ball recirculation limit are not checked.

The DC motor model ignores heating (resistance rises about 0.4 % per °C in copper), brush drop, and the driver’s current limit. Continuous torque is limited by temperature, not by this line — check the datasheet’s rated (continuous) current.

Servo ratings are stall torques at a stated voltage and are often optimistic.

Will it move? is a rigid-body estimate: no belt stretch, resonance or cutting forces beyond the "extra force" you enter. Stepper torque falls with speed; use the pull-out torque at your speed.