# Cyclogram — the calculator guides and the glossary, in full Source: https://cyclogram.app/calculations/ — a free axis and motor sizing calculator. The text below is the guide under each calculator page and every definition of the glossary. Each guide page also holds the live calculator, preset for its question. The calculations run in SI with g = 9.81 m/s². --- # Axis and motor sizing calculator URL: https://cyclogram.app/calculations/ Six steps from the move you need to the motor that makes it — and, at the end, the move as a bar for your cyclogram. A mass, a distance and a time give a first result; every field has a ? that says what it means and what is typical. No account, and nothing you type leaves this browser. ## What the six steps work out 1. The motion. The distance or angle and the move time; the dwell at the end, the move back to the start and the rest at home, each switched on or off; a trapezoidal or S-curve profile; the orientation, the guide friction and any external force. Out come the peak speed, acceleration and jerk — over the whole cycle. 2. The actuator and the load. A ball screw, a belt drive, a rack and pinion, a rotary table or a pneumatic cylinder, an optional gearbox, and the parts that move. Out come the travel per motor turn and the inertia the motor sees. 3. The motion, drawn. Position, speed, acceleration and motor speed over the cycle, and the mechanism's own limits — a screw's critical speed and D·N, the force in a belt or at a rack's teeth. A cylinder's stroke is drawn as its air allows. 4. The motor. A servo, a stepper, a BLDC motor or an AC induction motor on a drive: the torque in every phase, the RMS over the cycle, and four checks with their margins, drawn on the motor's speed–torque map; up to two more motors can stand beside it, judged on the same move, each by its own kind's rules, on one map and in one table. A cylinder has no motor: here it gets its force, each way, and its stroke time. 5. Regeneration. The energy that comes back while the axis brakes, against what the drive's bus can absorb — and the brake resistor when it cannot. 6. Send to Cyclogram. The move becomes a bar on your cycle diagram: the move out, the stay at the end, the move back. ## Guides by topic Each guide opens the same calculator, set up for its question, and works through the formulas with a full example underneath. - Ball screw motor sizingSpeed, reflected inertia, torque per phase, RMS, critical speed and D·N for a ball screw axis. - Belt drive motor sizingPulley and motor speed, reflected inertia, torque, belt force — and the gearbox a belt axis needs. - Rack and pinion motor sizingPinion and motor speed, the inertia the motor sees, torque, the force at the teeth — and the gearbox. - Rotary table motor sizingTable and motor speed, the inertia of discs and parts off the axis, the indexing torque — and the gearbox. - Pneumatic cylinder stroke timeForce out and in, the load ratio, the stroke time the air allows, and the air each cycle costs. - Move time and motion profilePeak speed, acceleration and jerk from a distance and a move time; trapezoid, triangle and S-curve. - Servo motor sizingPeak torque at speed, RMS over the cycle, maximum speed and inertia ratio, each with its margin. - Stepper motor sizingPull-out torque at speed over a safety factor, duty cycle, step rate and inertia ratio. - Inertia ratioThe load's inertia at the motor through a screw, a belt or a gearbox, and the ratio that fixes it. - Brake resistor sizingThe energy an axis returns while braking, what the drive's bus takes, and the resistor for the rest. - Ball screw critical speedHow fast a screw may turn for its root diameter, length and supports, and the nut's D·N limit. ## How the numbers are checked Every result has a “Show calculation” with the formula and your numbers in it. The mechanism torques come from calculation packages that are tested against Oriental Motor's published ball-screw, belt and index-table selection examples, within 0.5 %, and the cylinder's forces against a handbook's 6-bar table within 1 %. The torque the page samples over the cycle is held against the closed-form phase torques to 1 part in 100 000, and the default cases are worked by hand in the test suite — every number on these pages is recomputed there before each release. The page is in beta. The motors in the list are generic — typical sizes of each kind, not a maker's catalogue — and the envelope of a motor without its own curves is the generic shape of its kind. A cylinder's stroke time comes from an empirical model and is indicative once the load passes half the cylinder's force. ## Questions **Do I need an account to use the calculator?** No. Every calculation works without one. Only the last step — adding the move as a bar to your cyclogram — asks you to log in or sign up, because the cyclogram lives in the app. **Is anything I type sent anywhere?** No. The calculation, your saved calculations and your own motors stay in this browser. A file leaves it only when you download one. **Metric or imperial?** Both. The switch above the steps changes every field and every result; the calculation itself always runs in SI units. **Can I use my own motors?** Yes. Type one in by hand and save it, or drop catalogue files — JSON or CSV, one motor per file or per row — or a whole folder of them onto the Motor files box in step 4. **Which motors are in the list?** Generic motors of four kinds: servos from 50 W to 3 kW, steppers from NEMA 17 to NEMA 34, BLDC motors from 30 to 400 W, and AC induction motors from 0.37 to 4 kW. They are typical sizes, not a maker's catalogue. **How does the result get into a cyclogram?** Step 6 turns the move into a bar — the move out, the stay at the end position, the move back — and adds it to your cyclogram as a new lane, or downloads it as a library file. --- # Ball screw motor sizing URL: https://cyclogram.app/calculations/ball-screw-motor-sizing/ Enter the mass, the stroke and the move time; pick the lead and the screw. The calculator below gives the screw and motor speed, the inertia the motor sees and the torque in every phase, checks the screw's critical speed and D·N, and then tries a motor with margins. ## From table speed to motor speed A ball screw turns rotation into travel: one turn of the screw moves the nut by the lead p. A gearbox of ratio i between the motor and the screw divides that again, so one motor turn moves the table k: ``` k = p / i travel per motor turn n_screw = v / p · 60 rpm n_motor = n_screw · i ``` The default case moves 20 kg over 200 mm in 0.5 s, in equal thirds of accelerating, cruising and decelerating. That needs a peak speed of 0.6 m/s, and on a 10 mm lead the screw turns at 0.6 / 0.01 · 60 = 3600 rpm. ## The inertia the motor sees Seen from the motor, a mass moved by a screw behaves like a flywheel on the shaft: it is the mass on a drum of radius p/2π. The screw adds its own inertia — a solid steel cylinder unless you type the catalogue figure — and the coupling and a gearbox sit on the motor shaft as they are. ``` J_mass = m · (p / 2π)² / i² J_screw = m_screw · d² / 8 / i² m_screw = ρ · π/4 · d² · L, steel ρ = 7850 kg/m³ J_load = J_mass + J_screw + J_coupling + J_gearbox ``` For 20 kg on a 10 mm lead, J_mass = 20 · (0.01 / 2π)² = 5.07 × 10⁻⁵ kg·m². A Ø16 × 500 mm screw weighs 0.79 kg and adds 2.53 × 10⁻⁵ kg·m², so the motor sees 7.59 × 10⁻⁵ kg·m² — 2.92 times the rotor of the 400 W servo it is tried with. ## Torque in every phase The forces on the carriage — gravity along the axis, the guide friction, an external force — reach the motor through the lead and the efficiencies; a preloaded nut adds its own drag. While the axis accelerates, the inertias need torque on top, driven through the efficiency. While it decelerates, the load drives the motor back and the efficiency works the other way. ``` T_static = (m·g·sin α + μ·m·g·cos α + F) · p / (2π · η · i) + μ₀ · F₀ · p / (2π · i · η_t) α_motor = a / k · 2π T_accel = T_static + (J_mass + J_screw) · α / η + J_trans · α / η_t + J_rotor · α T_decel = T_static − ((J_mass + J_screw) · α · η + J_trans · α · η_t + J_rotor · α) T_dwell = T_gravity, or 0 with the brake holding T_rms = √( ∫ T² dt / t_cycle ) over the whole cycle: the moves, the dwell, the rest ``` Flat, with μ = 0.1 and η = 0.9, friction alone costs 0.0347 N·m. Accelerating at 3.6 m/s² the motor must give 0.284 N·m, cruising 0.0347 N·m, braking −0.179 N·m, and over the 1 s cycle the RMS is 0.138 N·m. Stand the same axis up vertically and gravity adds 0.347 N·m to every phase and to the dwell: the peak becomes 0.597 N·m and the RMS 0.378 N·m. ## The screw's own limits A long, thin screw whips at its critical speed, and the balls in the nut have a speed limit of their own, given as D·N — the nominal diameter times the screw speed. The default screw's critical speed is 8214 rpm with one fixed and one supported end; the page allows 80 % of it, 6572 rpm, and the screw turns at 3600. Its D·N is 57 600, under the usual 70 000. The critical speed guide works through both. ## Choosing the lead The lead is the screw's gear ratio. Doubling it from 10 to 20 mm halves the screw speed — 1800 rpm instead of 3600, and half the D·N — but every newton on the carriage costs twice the torque, and the moving mass looks four times heavier to the motor, because its inertia goes with p². A short lead suits heavy loads and fine positioning, a long one speed. Step 3 shows what the lead you pick does to the screw, step 4 what it does to the motor. ## Worked example | Quantity | By hand | Result | | --- | --- | --- | | Peak speed | 0.2 / (1/12 + 1/6 + 1/12) | 0.6 m/s | | Peak acceleration | 0.6 / (1/6) | 3.6 m/s² | | Screw and motor speed | 0.6 / 0.01 · 60 | 3600 rpm | | Load inertia at the motor | 5.07 × 10⁻⁵ + 2.53 × 10⁻⁵ | 7.59 × 10⁻⁵ kg·m² | | Inertia ratio | 7.59 × 10⁻⁵ / 2.6 × 10⁻⁵ | 2.92 | | Torque accelerating / cruising / braking | see above | 0.284 / 0.0347 / −0.179 N·m | | RMS torque over 1 s | √((0.284² + 0.0347² + 0.179²) / 6) | 0.138 N·m | | Peak torque the servo has at 3600 rpm | 3.82 · 3000 / 3600 | 3.18 N·m | | Margins: peak, RMS, speed, inertia ratio | (limit − required) / limit | 91 %, 89 %, 40 %, 71 % | | Braking energy per cycle | 0.179 · 377 / 2 · 1/6 | 5.61 J | The 400 W generic servo: 1.27 N·m rated, 3.82 N·m peak, 3000 rpm rated, 6000 rpm maximum, 2.6 × 10⁻⁵ kg·m² rotor. g = 9.81 m/s². ## Questions **What efficiency should I use for a ball screw?** About 0.9 for a ball screw. A trapezoidal lead screw is far lower, 0.3 to 0.5, and some of them hold a vertical load by friction alone. Enter the catalogue figure when you have it. **Does the nut preload add torque?** Yes. A preloaded nut drags with roughly μ₀ · F₀ · p / 2π at the screw, μ₀ about 0.1 to 0.3 by the catalogues, and the page adds it to every phase. Type the preload under “More about the screw”. **How do I size a ball screw for a vertical axis?** Choose Vertical in step 1. Gravity then appears in every phase and in the dwell, where the motor holds the load. With a motor brake switched on in step 4 the brake can hold the dwell instead, and the RMS torque falls. **What inertia ratio is acceptable for a ball screw axis?** Up to 10 for a standard servo is the usual rule. A ball screw on a direct-coupled servo rarely comes near it — the default case is 2.92 — but a heavy load on a long lead can. **What limits the speed of a ball screw?** Two things: the critical speed, from the root diameter, the length between the bearings and the end supports, and the D·N value the nut allows. Step 3 checks both. ## Sources - Oriental Motor, Technical Reference, selection calculations — the ball-screw load torque and the inertia of a mass driven by a screw. The page's test suite works its two published ball-screw examples and meets the printed answers within 0.5 %. - Ball-screw catalogues of NSK, THK and Bosch Rexroth — the critical-speed support factors and the D·N limit. --- # Belt drive motor sizing URL: https://cyclogram.app/calculations/belt-drive-motor-sizing/ A timing belt moves a carriage fast and far, but it shows the motor a large inertia. Enter the move and the pulleys; the calculator below gives the pulley and motor speed, the inertia the motor sees, the torque in every phase and the force in the belt — and the gearbox that makes the inertia ratio work. ## From carriage speed to motor speed A timing belt is a drum: one turn of the drive pulley moves the belt, and the carriage clamped to it, by the pitch circumference π·D. The pitch diameter is where the belt's cords run — the tooth count times the pitch, over π. ``` k = π · D / i travel per motor turn n_pulley = v / (π · D) · 60 rpm n_motor = n_pulley · i D = z · pitch / π z teeth ``` On Ø50 mm pulleys one turn moves 157 mm, so the default move's 0.6 m/s needs only 229 rpm. A belt axis turns its motor slowly, and that is where its trouble starts. ## The inertia the motor sees Everything the belt carries is a mass on a drum of radius D/2: the carriage, the payload and the belt itself, which moves at the same speed. The two pulleys add their own inertia; the page estimates them as solid cylinders of the pitch diameter and width, in aluminium or steel, unless you type the catalogue figure. ``` J_mass = (m + m_belt) · (D / 2)² / i² J_pulleys = 2 · m_pulley · D² / 8 / i² m_pulley = ρ · π/4 · D² · w J_load = J_mass + J_pulleys + J_coupling + J_gearbox ``` 20 kg and a 0.3 kg belt on Ø50 mm pulleys: J_mass = 20.3 · 0.025² = 1.27 × 10⁻² kg·m². Two solid aluminium pulleys 32 mm wide weigh 0.17 kg each and add 1.06 × 10⁻⁴ kg·m². The motor sees 1.28 × 10⁻² kg·m² — about 170 times what the same load looks like on a 10 mm ball screw. ## Torque, and the force in the belt The torques follow the same rule as a screw's, with the pulley radius in place of p/2π. The force in the belt is what the carriage needs — gravity, friction, the external force and the mass times the acceleration — and it is what the belt's catalogue rates. Belt tension does not reach the motor: it loads the bearings. ``` T_static = (m·g·sin α + μ·m·g·cos α + F) · D / (2 · η · i) T_accel = T_static + (J_mass + J_pulleys) · α / η + J_trans · α / η_t + J_rotor · α F_belt = m·g·sin α + μ·m·g·cos α + F ± (m + m_belt) · a ``` For the default move, friction costs 0.545 N·m at the motor; accelerating takes 2.60 N·m, cruising 0.545 N·m, braking −1.12 N·m, and the RMS over the cycle is 1.17 N·m. The belt carries 19.6 N of friction and 92.7 N at the peak. Type the belt's allowable force in step 2 and step 3 checks it. ## Why belt axes need a gearbox The same 400 W servo that drove the ball screw comfortably now drives 492 times its own rotor inertia. The torque is there — the peak needs 2.60 of the 3.82 N·m the motor has — but no servo loop settles a load like that cleanly. The load's inertia falls with the square of a gearbox's ratio, so the ratio that brings it down to the limit of 10 is ``` i = √( (J_load / J_rotor) / 10 ) = √( 492 / 10 ) ≈ 7.0 ``` The page suggests about 7.1 : 1. With a 7.1 : 1 gearbox the ratio is 9.76 — just under the limit, so the check reads Close — and an 8 : 1 gearbox brings it to 7.7. The motor then turns at 1627 rpm instead of 229, and the peak torque drops to 0.411 N·m. ## Worked example | Quantity | By hand | Result | | --- | --- | --- | | Travel per turn | π · 50 mm | 157 mm | | Pulley and motor speed | 0.6 / 0.157 · 60 | 229 rpm | | Moved mass and belt at the motor | 20.3 · 0.025² | 1.27 × 10⁻² kg·m² | | Both pulleys | 2 · 0.17 · 0.05² / 8 | 1.06 × 10⁻⁴ kg·m² | | Torque accelerating / cruising / braking | α = 3.6 / 0.025 = 144 rad/s² | 2.60 / 0.545 / −1.12 N·m | | RMS torque over 1 s | √((2.60² + 0.545² + 1.12²) / 6) | 1.17 N·m | | Belt force at the peak | 20.3 · 3.6 + 19.62 | 92.7 N | | Inertia ratio, direct drive | 1.28 × 10⁻² / 2.6 × 10⁻⁵ | 492 | ## Questions **Does belt tension add to the motor torque?** No. Pretension loads the bearings and stretches the belt; the motor sees only the forces that move the load, so it does not appear in the torque. **Which pulley diameter do I enter?** The pitch diameter, where the belt's tension cords run — not the tips of the teeth. It is the tooth count times the pitch over π: 30 teeth on a 5 mm pitch is 47.7 mm. **How do I find the belt's mass?** Belt catalogues give a mass per metre for each width; multiply it by the length of the whole loop. It is usually small next to the carriage, and 0 is a fair start. **Why does the page suggest a gearbox for my belt axis?** Because the inertia ratio is far over the limit. A belt reflects the load through the pulley radius, which is large next to a screw's lead, and the reflected inertia falls with the square of a gearbox ratio. **What is the allowable belt force?** The largest peripheral force the belt may carry, as its catalogue states it for the width, the pitch and the teeth in mesh. Step 3 compares the force at the peak of the move with it. ## Sources - Oriental Motor, Technical Reference, selection calculations — the belt-and-pulley load torque and inertia. The test suite works their belt-conveyor and belt-driven-table examples and meets the printed answers within 0.5 %. - Yaskawa, SigmaSelect help manual, the mechanism equations — the travel per motor turn πD/i. --- # Rack and pinion motor sizing URL: https://cyclogram.app/calculations/rack-and-pinion-motor-sizing/ A rack and pinion drives a long axis — a gantry, a transfer line — with the motor riding on the carriage, or a shorter one with the motor fixed and the rack moving. Enter the move, the arrangement, the module and the pinion's teeth; the calculator below gives the pinion and motor speed, the inertia the motor sees, the torque in every phase and the force at the teeth — and the gearbox the inertia ratio asks for. ## From module and teeth to motor speed The pinion rolls along the rack on its pitch circle, whose diameter is the module times the number of teeth. One turn of the pinion moves the carriage π·D; a gearbox in between divides that by its ratio. ``` D = m · z module × teeth k = π · D / i travel per motor turn n_pinion = v / (π · D) · 60 rpm n_motor = n_pinion · i ``` Module 2 with 20 teeth makes a Ø40 mm pinion. One turn moves 125.7 mm, and the default move's 0.6 m/s turns it at 286 rpm. ## Motor on the carriage, or the rack moving A rack-and-pinion axis is built one of two ways, and they move different masses. With the motor on the carriage, the rack is bolted to the frame and the pinion runs along it: the motor, the gearbox and their bracket ride with the load, so their mass moves too — step 2 asks for it. With the motor on the frame, the pinion stays where it is and pushes the rack, and the rack moves with the load: its own mass is what moves too, estimated from its length and face width unless you type the catalogue figure. ``` motor on the carriage: m = m_load + m_motor motor on the frame: m = m_load + m_rack m_rack = 7850 · w² · L ``` The default 20 kg load with 3 kg riding on the pinion makes 23 kg. The same axis with a fixed motor moves a 400 mm rack of 1.26 kg instead: 21.3 kg, and less torque. In exchange, a moving rack must be the stroke plus the pinion's mesh long, and it sweeps that length out past both ends of the axis. ## The inertia the motor sees Everything that moves is a mass on a drum of radius D/2. The pinion adds its own inertia; left blank, the page estimates it as a solid steel disc of the pitch diameter and the face width. ``` J_mass = m · (D / 2)² / i² J_pinion = m_p · D² / 8 / i² m_p = 7850 · π/4 · D² · w J_load = J_mass + J_pinion + J_coupling + J_gearbox ``` 23 kg on a Ø40 mm pinion: J_mass = 23 · 0.02² = 9.20 × 10⁻³ kg·m². A steel pinion 20 mm wide weighs 0.197 kg and adds 3.95 × 10⁻⁵ kg·m². The motor sees 9.24 × 10⁻³ kg·m². ## Torque, and the force at the teeth The torques follow the same rule as a belt's, with the pitch radius as the lever and η the mesh's and the gearbox's efficiency together. The force at the teeth is what the carriage needs — gravity, friction, the external force and the mass times the acceleration — and it is what the rack's catalogue rates for the module and the face width. ``` T_static = (m·g·sin α + μ·m·g·cos α + F) · D / (2 · η · i) T_accel = T_static + (J_mass + J_pinion) · α / η + J_trans · α / η_t + J_rotor · α F_teeth = m·g·sin α + μ·m·g·cos α + F ± m · a ``` For the default move with the motor on the carriage, friction costs 0.501 N·m at the motor; accelerating takes 2.35 N·m, cruising 0.501 N·m, braking −1.00 N·m, and the RMS over the cycle is 1.06 N·m. The teeth carry 22.6 N of friction and 105 N at the peak. Type the rack's allowable force in step 2 and step 3 checks it. ## The gearbox Straight on the pinion, the 400 W servo drives 355 times its own rotor inertia — less than a belt drive's, for the smaller diameter, but far over the limit of 10. The reflected inertia falls with the square of a gearbox's ratio: ``` i = √( (J_load / J_rotor) / 10 ) = √( 355 / 10 ) ≈ 6.0 ``` The page suggests about 6.0 : 1. With it the ratio is 9.87 — just under the limit, so the check reads Close — and 6.3 : 1 brings it to 8.95. Through 6 : 1 the motor turns at 1719 rpm and the peak torque drops to 0.440 N·m. With the motor fixed and the rack moving, the lighter 21.3 kg gives a ratio of 329 direct. ## Choosing the pinion The force at the teeth does not depend on the pinion: it is the carriage's force. The pinion's size changes what that force costs at the motor — the torque rises with D, the reflected inertia with D², the speed falls with 1/D. So pick the module for the force, from the rack's catalogue, then the fewest teeth that still run smoothly: below about 17, a standard 20° pinion undercuts its own teeth. ## Worked example | Quantity | By hand | Result | | --- | --- | --- | | Pitch diameter | 2 mm · 20 | 40 mm | | Travel per turn | π · 40 mm | 125.7 mm | | Pinion and motor speed | 0.6 / 0.1257 · 60 | 286 rpm | | Moving mass | 20 kg load + 3 kg motor and gearbox | 23 kg | | Moving mass at the motor | 23 · 0.02² | 9.20 × 10⁻³ kg·m² | | Pinion | 0.197 · 0.04² / 8 | 3.95 × 10⁻⁵ kg·m² | | Torque accelerating / cruising / braking | α = 3.6 / 0.02 = 180 rad/s² | 2.35 / 0.501 / −1.00 N·m | | RMS torque over 1 s | √((2.35² + 0.501² + 1.00²) / 6) | 1.06 N·m | | Force at the teeth at the peak | 23 · 3.6 + 22.56 | 105 N | | Inertia ratio, direct | 9.24 × 10⁻³ / 2.6 × 10⁻⁵ | 355 | ## Questions **Is the module the same as the pitch?** No. The module is the pitch diameter over the number of teeth, in millimetres; the pitch along the rack, tooth to tooth, is π times the module — 6.28 mm for module 2. **The motor rides on the carriage — does its mass count?** Yes. Choose “On the carriage” in step 2 and type what rides with the pinion — the motor, the gearbox and their bracket. The page adds it to the moving mass, to the friction and, on a climbing axis, to the weight the motor lifts. **Motor on the carriage or on the frame — which is better?** On the carriage, the rack can run the whole length of the machine and several carriages can share it, but the motor's cable travels with it and its mass moves. On the frame, the cable stays put and only the rack moves, but the rack must be the stroke plus the mesh long and sweeps out past both ends — usually the choice for shorter strokes. **How do I find the pinion's inertia?** From the pinion's catalogue, about its own axis. Left blank, the page takes a solid steel disc of the pitch diameter and the face width — close enough for sizing. **What is the allowable force at the teeth?** The largest tangential force the rack and pinion may carry, from the catalogue for the module, the face width, the quality and the hardening. Step 3 compares the force at the peak of the move with it. **Why does a smaller pinion help the motor?** The torque for the same carriage force falls with the diameter and the inertia the motor sees with its square. The motor turns faster in exchange, which a servo usually welcomes. ## Sources - Oriental Motor, Technical Reference, selection calculations — the rack-and-pinion load torque and inertia, the same lever as a belt's pulley. - ISO 54 — the series of modules for cylindrical gears. - The involute gear's undercut limit, z_min = 2 / sin²α, about 17 teeth at a 20° pressure angle. --- # Rotary table motor sizing URL: https://cyclogram.app/calculations/rotary-table-motor-sizing/ An index table turns a fixture through a fixed angle, again and again — and nearly all of its load is inertia. Enter the angle, the time and what turns; the calculator below gives the table and motor speed, the inertia the motor sees, the torque in every phase — and the gearbox that brings the inertia ratio into range. ## The angle, the time and the speed A rotary axis moves an angle instead of a distance; the profile is the same — accelerate, cruise, decelerate, dwell — in degrees. The table sits on the motor or behind a gearbox, and one motor turn turns it 360° / i. ``` ω_peak = θ / (t_move · (1 − (f_a + f_d) / 2)) rad/s n_table = ω_peak / 2π · 60 rpm n_motor = n_table · i α = ω_peak / t_accel ``` 90° in 0.5 s, in thirds, peaks at 4.71 rad/s — 45 rpm at the table — and accelerates at 28.3 rad/s². ## What turns: discs, and parts off the axis Almost everything on a table is a disc or close to one. A disc about its own centre is m·d²/8; a part whose centre sits a distance r from the axis adds m·r² on top — the parallel-axis rule — and that term is usually the larger. Step 2 takes each disc with its distance from the axis. ``` J_disc = m · d² / 8 about its own centre J_offset = m · d² / 8 + m · r² centre at r from the axis J_load = Σ J / i² ``` A 1.5 kg fixture Ø80 mm sitting 120 mm out: 1.5 · 0.08² / 8 = 1.2 × 10⁻³ kg·m² about its own centre, plus 1.5 · 0.12² = 2.16 × 10⁻² for where it sits — 2.28 × 10⁻² kg·m² in all, the offset eighteen times the disc. ## Torque A table's torque is almost all inertia: a level table's weight loads its bearings, not the motor. The page adds the bearings' friction torque and any external torque, both at the table, divided by the efficiencies and the gear ratio. ``` T_static = (T_external + T_friction) / (η · i) α_motor = α · i T_accel = T_static + J_load · α_motor / η + J_trans · α_motor / η_t + J_rotor · α_motor ``` The default table, 0.01 kg·m² straight on the 400 W servo: 0.283 N·m while accelerating, −0.283 N·m braking, nothing at constant speed, and 0.164 N·m RMS over the 1 s cycle. ## The gearbox Direct, the servo drives 385 times its own rotor inertia. A rotary table nearly always wants a reducer, and the reflected inertia falls with the square of its ratio: ``` i = √( (J_load / J_rotor) / 10 ) = √( 385 / 10 ) ≈ 6.2 ``` The page suggests about 6.3 : 1, which brings the ratio to 9.69 — Close — and 7 : 1 to 7.85. Through 6.3 : 1 the motor turns at 284 rpm and the peak falls to 0.0519 N·m: the gearbox, not the motor, now carries the table. ## Worked example | Quantity | By hand | Result | | --- | --- | --- | | Peak speed | (π/2) / (0.5 · 2/3) | 4.71 rad/s, 45 rpm | | Acceleration | 4.71 / (0.5 / 3) | 28.3 rad/s² | | Torque accelerating / braking | (0.01 + 2.6 × 10⁻⁵) · 28.3 | 0.283 / −0.283 N·m | | RMS torque over 1 s | 0.283 · √(2/6) | 0.164 N·m | | Inertia ratio, direct | 0.01 / 2.6 × 10⁻⁵ | 385 | | Inertia ratio through 6.3 : 1 | 0.01 / 6.3² / 2.6 × 10⁻⁵ | 9.69 | ## Questions **Does the weight of the table matter?** On a level table, only through its inertia: the weight loads the bearings, not the motor. A table on a tilted axis is out of balance when its load is — type that as an external torque in step 1. **How do I enter the parts around the table?** As discs, each with its mass, its diameter and its distance from the axis — or as one disc carrying the mass of all of them at the same distance. For any other shape, type its inertia from the CAD model. **Which angle and time for an indexing table?** The index angle is 360° over the number of stations: eight stations is 45°. The move time is the part of the machine's cycle the table may take to index; the rest is the dwell while the stations work. **Why does a rotary table need a gearbox?** Because nearly all of its load is inertia, and the inertia the motor sees falls with the square of the ratio. A few kilograms of fixtures well out from the axis easily show a small servo hundreds of times its rotor inertia. **What is the table drive efficiency?** For a table with its own drive inside — a worm, a belt or a cam — the share of the torque that reaches the table. A table straight on the gearbox has nothing in between: leave it at 1. ## Sources - Oriental Motor, Technical Reference, selection calculations — the index-table example. The test suite works it and meets the printed load inertia and acceleration torque within 0.5 %. - The parallel-axis theorem — J = J_centre + m · r² for a part off the axis. --- # Pneumatic cylinder stroke time URL: https://cyclogram.app/calculations/pneumatic-cylinder-stroke-time/ A pneumatic cylinder has no motor to size: the air it gets sets its speed, so the stroke time is the result. Enter the bore, the pressure and the air flow; the calculator below gives the force each way, the load ratio, the stroke time against the move time you need and the air each cycle costs — and the stroke as a bar for your cyclogram. ## The force, each way The air pushes on the piston's whole area to extend, and only on the ring around the rod to retract. Seal friction takes about 12 % of it; the page keeps 0.88. ``` A_out = π · d² / 4 A_in = π · (d² − d_rod²) / 4 F = p · A · 0.88 p in N/mm²: bar / 10 ``` A Ø32 cylinder with its usual 12 mm rod, at 6 bar: 425 N out and 365 N in. ## The load ratio The load ratio is the force the load needs over the force the cylinder makes. Under 0.5 the air sets the speed and the load hardly matters; above it the stroke slows, above 0.8 it crawls and can stall, and at 1 it does not move. The page stretches the time by 1 / (1 − ratio) — an empirical model, indicative above 0.5. ``` F_load = m·g·sin α + μ·m·g·cos α + F_ext on the extend ratio = F_load / F ``` The default 20 kg on a level guide with μ 0.1 asks 19.6 N of the cylinder: a load ratio of 0.046. Stood vertical, lifting the same 20 kg, it is 0.462; 30 kg is 0.693 — Close. ## The stroke time A stroke takes the air that fills the barrel — and the tube from the valve — at the supply pressure, counted as free air at atmospheric pressure. The flow the valve and its fittings deliver sets how long that takes; the valve's own delay and the end cushioning add to it. ``` V_free = (A · s + V_tube) · (p + 1.013) / 1.013 V_tube = π · d_t² / 4 · L t_flow = V_free / (Q · k) t = t_flow / (1 − ratio) + t_valve + t_cushion ``` The default stroke, 200 mm on Ø32 at 6 bar with 1 m of 4 mm tube, takes 1.20 l of free air; at 100 l/min that is 0.720 s. With the load ratio, the 0.08 s valve delay and 0.05 s of cushioning the extend takes 0.885 s — inside the 1 s this page opens with. The retract, on the smaller rod-side area, takes 0.792 s. ## Making it faster Cut the move time to 0.5 s and the same cylinder is too slow. More air flow is the cure — a bigger valve, a wider or shorter tube, the flow control opened: at 250 l/min the stroke takes 0.432 s. A smaller bore needs less air, if its force still covers the load. A bigger bore helps only when the load ratio is the trouble: lifting 30 kg, Ø40 instead of Ø32 brings the ratio from 0.693 to 0.444 and the stroke from 2.48 s to 2.10 s, though it takes 3.38 l of air per cycle instead of 2.24 l. ## The air it costs Each cycle fills the barrel on both sides of the piston, and the tube each time: 2.24 l of free air for the default. Back to back with its 0.5 s dwell, that is 61.9 l/min — the figure for the compressor, though on a machine the cylinder also waits for the rest of the cycle, which lowers it. ## Worked example | Quantity | By hand | Result | | --- | --- | --- | | Area out / in | π · 32² / 4, π · (32² − 12²) / 4 | 804 / 691 mm² | | Force out / in at 6 bar | 0.6 N/mm² · A · 0.88 | 425 / 365 N | | Load ratio, extending | 19.62 / 424.6 | 0.046 | | Free air, extending | (160.8 + 12.6) cm³ · 7.013 / 1.013 | 1.20 l | | Flow time | 1.20 l / (100 / 60) l/s | 0.720 s | | Extend | 0.720 / (1 − 0.046) + 0.08 + 0.05 | 0.885 s | | Retract | the rod side, load ratio 0.054 | 0.792 s | | Air per cycle | both strokes | 2.24 l | ## Questions **Which air flow do I enter?** The flow that actually reaches the cylinder, in litres of free air per minute — the least of what the valve, the fittings and the tube let through. A valve's nominal flow is an upper bound; long thin tubes lower it. **Why is the retract faster?** The rod takes up part of the barrel, so the rod side holds less air to fill. Its force is smaller too, but while the load is light the air, not the force, sets the time. **What is a good load ratio?** Under 0.5 for a quick, steady stroke. Between 0.5 and 0.8 is acceptable for a clamp or a press that only has to arrive; above 0.8 the stroke crawls and can stall. A vertical cylinder lifting a load needs the margin most. **Does the page model the tube?** Its volume, yes: the tube fills with every stroke, which costs air and time. Its flow resistance, no — fold that into the air flow you enter. **How does the cylinder become a bar on a cyclogram?** Step 6 makes the extend the bar's move out, the dwell its stay, and the cylinder's own computed retract its move back. ## Sources - Mechanical and Metal Trades Handbook (Europa-Lehrmittel), pneumatic cylinders — the piston forces at 6 bar. The page's forces meet its table within 1 %. - ISO 15552 and ISO 6432 — the bore series and the usual rod for each bore. - The 1 / (1 − ratio) slowdown is an empirical model, indicative above a ratio of 0.5; real stroke times scatter with pressure, temperature and wear. --- # Move time, peak speed and acceleration URL: https://cyclogram.app/calculations/move-time/ Type the distance, the move time and how the move splits into accelerating, cruising and decelerating. The calculator below gives the peak speed, acceleration and jerk, draws the profile, and carries it on to the mechanism and the motor. ## The trapezoid A trapezoidal profile accelerates at a constant rate, cruises, and decelerates. With the ramps given as fractions of the move time T, the distance is the area under the speed curve, and the peak speed follows from it: ``` t_a = f_a · T t_d = f_d · T t_c = T − t_a − t_d s = v · (t_a/2 + t_c + t_d/2) → v = s / (t_a/2 + t_c + t_d/2) a = v / t_a d = v / t_d ``` 200 mm in 0.5 s in equal thirds: v = 0.2 / (1/12 + 1/6 + 1/12) = 0.6 m/s, and a = 0.6 / (1/6) = 3.6 m/s², a little over a third of g. ## Why a third, a third, a third Spending the whole move on the ramps — a triangle — asks for the least acceleration but the most speed: 3.2 m/s² and 0.8 m/s for the same move. Short ramps cost acceleration, long ramps cost speed. For a load that is mostly inertia, the peak power goes with a·v and the RMS torque with a·√(ramp time), and with equal ramps of a fraction f each, both are smallest at f = 1/3: ``` a = s / (T² · f · (1 − f)) v = s / (T · (1 − f)) P_peak ∝ a · v ∝ 1 / (f · (1 − f)²) smallest at f = 1/3 T_rms² ∝ a² · 2f ∝ 1 / (f · (1 − f)²) smallest at f = 1/3 ``` The calculator agrees: the default ball-screw axis needs an RMS torque of 0.138 N·m in thirds and 0.150 N·m as a triangle. The triangle has a practical cost too: its 0.8 m/s turns the 10 mm screw at 4800 rpm, and the D·N climbs to 76 800 — over the usual 70 000, which step 3 flags. ## The S-curve An S-curve limits the jerk — the rate at which the acceleration changes — by ramping the acceleration up and down instead of switching it. Its ramps stay symmetric in time, so the area under each is still half the speed times the ramp time, and the peak speed is the trapezoid's. The price is a higher peak acceleration, set by the jerk fraction r, the share of each ramp spent changing the acceleration: ``` v = s / (t_a/2 + t_c + t_d/2) as the trapezoid a = v / (t_a · (1 − r)) j = a / (r · t_a) ``` With r = 0.3 the default move peaks at 5.14 m/s² instead of 3.6, with a jerk of 103 m/s³. On the ball screw the peak torque rises from 0.284 to 0.391 N·m, but it arrives later in the ramp, at 2829 rpm, where the servo still has its full peak torque. ## The dwell The time the axis stands counts as well. It is part of the RMS torque, which is averaged over the whole cycle — the move out, the dwell, the move back when the axis returns, the rest at home — and it spreads the braking energy of step 5 over a longer cycle. A vertical axis holds its weight through the dwell unless a brake does. ## The same move, three ways | 200 mm in 0.5 s, 10 mm lead | Thirds | Triangle | S-curve, r = 0.3 | | --- | --- | --- | --- | | Peak speed | 0.6 m/s | 0.8 m/s | 0.6 m/s | | Peak acceleration | 3.6 m/s² | 3.2 m/s² | 5.14 m/s² | | Jerk | — | — | 103 m/s³ | | Motor speed | 3600 rpm | 4800 rpm | 3600 rpm | | Peak torque | 0.284 N·m | 0.257 N·m | 0.391 N·m | | RMS torque | 0.138 N·m | 0.150 N·m | 0.152 N·m | ## Questions **Trapezoid or S-curve?** A trapezoid is the usual sizing assumption and asks for the least acceleration in a given move time. An S-curve is gentler on the mechanism and the load — less vibration, less overshoot — at the cost of a higher peak acceleration and torque. Size with the profile your controller will actually run. **How should I split the move?** Equal thirds is the usual start and, for an inertia load, the cheapest in peak power and RMS torque. Shorter ramps lower the acceleration you can get away with only by raising the speed; a triangle (50 / 50) goes furthest that way. **What jerk fraction should I use?** 0.5 is a pure S with no constant-acceleration part, 0.2 to 0.3 a typical smoothing, and 0 would be a trapezoid. **Does the move time include the ramps?** Yes. The move time runs from standstill to standstill — it is the time the bar takes on the cyclogram. --- # Servo motor sizing: peak, RMS, speed and inertia ratio URL: https://cyclogram.app/calculations/motor-check/ Pick a motor and the calculator below checks it against the move: the peak torque at the speed where it happens, the RMS over the whole cycle, the top speed and the inertia ratio — each with its margin, drawn on the motor's speed–torque map. Steppers, BLDC and AC induction motors get the checks that fit them. Set up to two more motors beside the first to compare them on one map and in one table; every chart zooms and pans. ## What a servo can give A servo's catalogue gives a rated torque, which it holds for ever without overheating, and a peak torque, which it gives for a short while. Both hold up to the rated speed; above it the motor runs at constant power and the torque falls, to nothing at the maximum speed. Without the motor's own curves, the page draws that generic envelope: ``` T_cont(n) = T_rated n ≤ n_rated T_cont(n) = T_rated · n_rated / n n_rated < n ≤ n_max T_int(n) = the same with T_peak ``` The default move peaks at 3600 rpm on a 3000 rpm servo, so the 400 W generic motor's 3.82 N·m peak torque is derated to 3.82 · 3000 / 3600 = 3.18 N·m where it is needed. ## The four checks | Check | Against | Required | Limit | Margin | | --- | --- | --- | --- | --- | | Peak torque | the intermittent torque at the speed of the peak | 0.284 N·m | 3.18 N·m | 91 % | | RMS torque | the rated torque | 0.138 N·m | 1.27 N·m | 89 % | | Max speed | the maximum speed | 3600 rpm | 6000 rpm | 40 % | | Inertia ratio | the limit, 10 by default | 2.92 | 10 | 71 % | ``` margin = (limit − required) / limit T_rms = √( Σ T_i² · t_i / (t_move + t_dwell) ) inertia ratio = J_load / J_rotor ``` The margin is the share of the limit still unused. Below zero the row fails; below 10 % it reads Close — you can set that threshold, and the inertia-ratio limit, under Limits in step 4. ## When the inertia ratio fails The smallest generic servo, 50 W, has the torque for the default move — 42 % margin on the peak — but its rotor is so light that the load is 30.4 times its inertia. The page suggests the gearbox that would bring the ratio to 10, i = √(30.4 / 10), about 1.8 : 1. The inertia ratio guide works through it. ## Steppers, BLDC and AC induction motors - Stepper — sized on its pull-out torque at speed over a safety factor, with the duty cycle and the step rate checked too. See the stepper guide. - BLDC motor — a rated torque and a limited-duty torque, both constant over the speed range; the load inertia is checked against the maker's permissible figure when you give it. - AC induction motor on a drive — the rated torque up to the base speed, constant power beyond, and a breakdown torque that falls with the square of the speed in field weakening. Its inertia ratio is shown as a caution only, as there is no servo loop to tune. ## Questions **Why check the RMS torque?** Because heat decides whether a motor survives the cycle. The RMS torque over the whole cycle — the moves, the dwell and the rest — is the steady torque that would warm the motor as much, and it must stay under the rated torque. **What margin should I keep?** The page reads a margin under 10 % as Close. Many designers keep 20 to 30 % on the peak and the RMS for unknown friction, wear and a faster cycle later; set your own threshold under Limits in step 4. **What if the move runs above the rated speed?** Then the torque available is lower: the page derates the peak and rated torque at constant power above the rated speed, and checks the peak against the derated value at the speed where it happens. **Can I use a motor's real speed–torque curves?** Yes. A motor brought in from a file can carry its own curves as speed–torque pairs; the page interpolates them instead of the generic envelope. **Can I load a whole motor catalogue?** Yes. Drop JSON or CSV files — or a folder of them — onto the Motor files box in step 4. Common column names and units are understood, and the motors join My motors in this browser. --- # Stepper motor sizing URL: https://cyclogram.app/calculations/stepper-motor-sizing/ A stepper is sized on its pull-out torque, not its holding torque, with a safety factor — and it runs warm even standing still. The calculator below, set up with a NEMA 23 stepper, checks all of it for your move: pull-out torque at speed, duty cycle, step rate and inertia ratio. ## Holding torque is not what it gives The holding torque on the catalogue's first line is what a stepper holds at standstill. Turning, it gives its pull-out torque, which starts near the holding torque and falls away with speed as the windings run out of time to fill with current. Without the motor's own curve the page takes it flat up to a corner speed and falling as 1/n beyond, and lets the move use it over a safety factor: ``` T_pull(n) = T_hold n ≤ n_c T_pull(n) = T_hold · n_c / n n_c < n ≤ n_max usable = T_pull / SF SF = 2 by default step rate = n / 60 · 360 / θ_step steps/s ``` ## A worked example A generic NEMA 23 stepper — 1.2 N·m holding torque, the torque holding to 300 rpm, a 1500 rpm maximum — on the default 10 mm ball screw, moving 100 mm in 1 s in thirds with a 1.5 s dwell: - The peak speed is 0.15 m/s, so the motor turns at 900 rpm, 3000 steps/s at 1.8° a step. - At 900 rpm the pull-out torque is 1.2 · 300 / 900 = 0.4 N·m; over the safety factor of 2 the move may use 0.2 N·m. - The move needs 0.0665 N·m at its peak, a margin of 67 %. - The motor moves for 40 % of the 2.5 s cycle, under the 50 % a stepper at full current tolerates. - The load is 2.71 times the rotor's inertia. ## Where steppers stop The same motor on the default move — 200 mm in 0.5 s, 3600 rpm at the motor — fails twice. 3600 rpm is past its 1500 rpm maximum, and at 1500 rpm its usable torque is 0.12 N·m against the 0.289 N·m the move needs. Above about 1000 rpm a stepper has little torque left; a servo or a closed-loop stepper suits such a move better. ## The duty cycle A stepper draws its full current standing still, so it heats over the whole cycle, not just while it moves. The page warns — marked Hot — when the motor moves for more than half the cycle: lengthen the dwell, or let the driver reduce the standstill current. ## Questions **What safety factor should a stepper have?** 2 is the usual rule, 1.5 where the load and the friction are well known. Open loop forgives nothing: a stall is a lost position, found only at the next homing. **Does microstepping change the torque?** It smooths the motion and refines the resolution, but it does not add torque. Size on the pull-out torque your driver's curve gives. **Can I enter the pull-out curve?** Yes. A motor file can carry the pull-out curve as speed–torque pairs, and the page then uses it instead of the generic shape. **Why does the driver's supply voltage matter?** A higher supply fills the windings faster, so the pull-out torque holds to a higher speed. Type the corner speed for your driver and voltage, or use its curve. **What inertia ratio suits a stepper?** About 5 when it drives the load directly and 10 through a gearbox is the usual rule; beyond it the motor tends to miss steps on the ramp. ## Sources - Oriental Motor, Technical Reference, stepping motor selection — the safety factor on the pull-out torque and the duty limit of a standard stepper at full current. --- # Inertia ratio: load to rotor URL: https://cyclogram.app/calculations/inertia-ratio/ How heavy the load looks from the motor shaft, against the motor's own rotor — through a ball screw, a belt or a gearbox — and the gear ratio that brings it under the limit. The calculator below is set up with a belt drive, where the ratio matters most. ## Reflecting the load to the motor A load does not reach the motor as kilograms but as an inertia on the shaft, set by how far it moves per motor turn. Everything behind a gearbox falls with the square of its ratio: | Mechanism | Load inertia at the motor | | --- | --- | | Ball screw, lead p | m · (p / 2π)² / i² + J_screw / i² | | Belt or rack, pitch diameter D | (m + m_belt) · (D / 2)² / i² + J_pulleys / i² | | Rotary table | J_table / i² | | On the motor shaft | J_coupling, J_gearbox as they are | ## Why the ratio matters A servo controls the load through its own rotor. When the load is many times heavier, the loop has to be tuned soft to stay stable, and the axis settles slowly and overshoots. Up to 10 times the rotor is the usual rule for a standard servo, 5 for a very dynamic axis, up to 30 with a stiff coupling and careful tuning. A stepper follows its ramp with about 5 times direct and 10 through a gearbox. A BLDC motor's maker states a permissible load inertia instead, and an induction motor on a drive has no loop to tune — there the ratio is only a caution. ## Finding the gear ratio ``` ratio with a gearbox = ratio without / i² i needed = i_now · √( ratio / limit ) ``` - The default ball screw on the 400 W servo: 2.92. Nothing to do. - The same screw on the 50 W servo: 30.4, so i = √(30.4 / 10), about 1.8 : 1. - The default belt drive on the 400 W servo: 492. The page suggests about 7.1 : 1; with it the ratio is 9.76, just under the limit and marked Close. An 8 : 1 gearbox brings it to 7.7. ## What a gearbox costs A gearbox multiplies the motor speed by its ratio — the belt behind 7.1 : 1 turns its motor at 1627 rpm instead of 229 — so check the maximum speed; it loses a little to its efficiency; and it adds its own inertia at the motor shaft, which you can type in step 2. In return the torque the motor must give falls with the ratio: the belt's peak drops to 0.411 N·m. ## Questions **Is an inertia ratio of 10 a hard limit?** No. It is the usual rule for a standard servo tuned without effort. Very dynamic axes want 5 or less; a stiff mechanism with careful tuning can run 30. Set your own limit under Limits in step 4. **Does the gearbox's own inertia count?** Yes — it sits on the motor shaft, so it counts in full. Type it with the gearbox in step 2. The suggested ratio counts the rotor only, so check the ratio again with the gearbox in. **Why does the ratio fall with the square of the gear ratio?** Because the load turns i times slower than the motor: its speed falls by i and its kinetic energy, which goes with speed squared, by i². The inertia the motor sees falls by the same i². **Why not take a bigger motor instead of a gearbox?** A heavier rotor lowers the ratio too, but it buys torque the move may not need. A gearbox also lowers the torque the motor must give, so a small motor with a gearbox is often the cheaper answer. --- # Brake resistor sizing: regenerative energy URL: https://cyclogram.app/calculations/brake-resistor-sizing/ When an axis brakes, the motor becomes a generator and pushes energy back into the drive. The calculator below works out how much per cycle, from the motion profile of step 1, compares it with what the drive's DC bus can absorb, and sizes the brake resistor when it cannot. ## The energy that comes back Wherever the motor's torque and its speed point opposite ways, the load is driving the motor. The page integrates that power over the cycle and takes off the motor's and the drive's losses: ``` E_mech = ∫ max(0, −T(t) · ω(t)) dt over the whole cycle E_net = E_mech · η_regen η_regen ≈ 0.9 ``` The default ball-screw axis brakes with 0.179 N·m from 3600 rpm (377 rad/s) to standstill in 1/6 s: E_mech = 0.179 · 377 / 2 · 1/6 = 5.61 J, of which 5.05 J reaches the bus. The peak braking power, at the start of the ramp, is 60.6 W after the losses. ## What the bus can take A drive absorbs braking energy in its DC-bus capacitors until the voltage reaches the brake chopper's threshold. The data sheet states the energy, or it follows from the capacitance: ``` E_abs = ½ · C · (V_max² − V_bus²) ``` 1000 µF charged from 320 V to 390 V takes ½ · 0.001 · (390² − 320²) = 24.9 J — more than the default axis returns, so it needs no resistor. ## When a resistor is needed If the bus can take only 0.5 J, the remaining 4.55 J of every cycle must go to a resistor. Spread over the 1 s cycle that is an average of 4.55 W, and the resistor's short-time rating must cover the 60.6 W peak. ``` P_avg = (E_net − E_abs) / t_cycle P_peak = max(−T · ω) · η_regen ``` ## Vertical axes Which way the axis moves decides a great deal. Climbing, gravity brakes the load for the motor: the default axis stood up vertically returns nothing at all, because even its deceleration needs a positive torque, 0.134 N·m. Coming down, gravity drives the motor through the whole move, and the energy grows with the height. ## Based on the motion profile The energy is worked out for the cycle step 1 defines: the move out, the dwell, the move back when it is switched on, and the rest at home. A longer dwell or rest lowers the resistor's average power; a move back adds its own braking at the end, and on a vertical axis the whole way down — the weight the motor lifted comes back through the drive. A belt axis brakes the same way: the default belt drive returns 2.23 J per cycle. ## Questions **When do I need a brake resistor?** When one cycle returns more energy than the drive's bus can absorb. Heavy loads, high speeds, short ramps and descending vertical axes are the usual causes. **What is the chopper threshold?** The bus voltage at which the drive switches its brake resistor in, a little under its overvoltage trip. Between the normal bus voltage and that threshold, the capacitors store the braking energy. **How do I rate the resistor?** For the average power over the cycle, and with a short-time rating above the peak braking power. Both are in step 5 when a resistor is needed. **Does the regeneration efficiency matter?** Yes. The motor's winding losses and the drive's losses take part of the braking energy before it reaches the bus; 0.9 of it arriving is a common assumption, and you can set it in step 5. --- # Ball screw critical speed and D·N URL: https://cyclogram.app/calculations/ball-screw-critical-speed/ How fast a ball screw may turn before it whips — from its root diameter, the length between its bearings and how its ends are held — and the D·N limit of its nut. Step 3 of the calculator below checks both for your move. ## The critical speed A spinning screw is a shaft between bearings with a natural frequency of bending. Turned near it, it whips. The screw catalogues give the speed as ``` n_c = f · d_r / L² · 10⁷ rpm, with d_r and L in mm allowed = 0.8 · n_c ``` where d_r is the root diameter, L the length between the bearings, and f a factor for how the ends are held. For a Ø16 mm screw — root 13.6 mm — 500 mm between bearings: | Ends | f | n_c | | --- | --- | --- | | fixed – free | 3.4 | 1850 rpm | | supported – supported | 9.7 | 5277 rpm | | fixed – supported | 15.1 | 8214 rpm | | fixed – fixed | 21.9 | 11 914 rpm | The root diameter is the screw's core under the thread; the page takes 0.85 times the nominal diameter unless you type the catalogue's figure. ## The longest screw for a speed ``` L_max = √( f · d_r · 10⁷ · 0.8 / n ) ``` At 3600 rpm the Ø16 mm screw may be 676 mm between bearings with one fixed and one supported end, or 814 mm with both ends fixed. At 800 mm, fixed – supported, its critical speed falls to 3209 rpm, and the page flags the move. ## D·N — the nut's own limit ``` D·N = d · n_screw ≤ about 70 000 for a standard screw ``` The balls circulating in the nut have a speed limit of their own. 16 mm at 3600 rpm is 57 600. Stretch the same move into a triangle profile and the screw reaches 4800 rpm and a D·N of 76 800 — over. High-speed screw series allow more; take the figure from the catalogue. ## When the screw is too fast - Shorten the span between the bearings, or stiffen the ends — fixed – fixed raises the critical speed by 2.3 times over supported – supported. - Take a larger diameter: the critical speed rises with it, though so do the D·N and the screw's inertia. - Take a longer lead: the screw turns slower for the same table speed. - For long, fast strokes, a belt drive has no critical speed at all. ## Questions **Why only 80 % of the critical speed?** Real supports are never as stiff as the formula assumes, and the screw must not run close to its resonance. 80 % is the common catalogue allowance. **What root diameter should I enter?** The catalogue's root (core) diameter of the thread. Without it, the page takes 0.85 times the nominal diameter, a typical value. **Which end supports does my screw have?** Fixed means a pair of angular-contact bearings that hold the shaft's angle; supported means a single bearing that lets it tilt; free means no bearing at that end. **Which length do I enter?** The length between the bearings. For a fixed – free screw, catalogues measure from the bearing to the nut at its farthest position. ## Sources - Ball-screw catalogues of NSK, THK and Bosch Rexroth — the critical-speed formula, the support factors and the D·N limit. --- # The glossary URL: https://cyclogram.app/calculations/glossary/ ## Motion profile terms URL: https://cyclogram.app/calculations/glossary/motion/ ### Linear or rotary axis Whether the thing that moves travels along a line (a carriage) or turns through an angle (a table). Typical: linear: a carriage on a screw, belt or rack; rotary: an indexing table, a turning shaft. It decides the words on every field below: a stroke in millimetres or an angle in degrees, a mass or an inertia. ### Moving mass Everything the axis carries along: the carriage, the workpiece, the fixtures and the nut. Unit: kg. Typical: 2 – 50 kg for a small axis, 100 – 500 kg for a machine table. Step 2 lets you list the parts one by one; this field is the first of them. ### Load inertia How hard the load is to spin up: its mass times the square of how far that mass sits from the axis. Unit: kg·m². Typical: 0.001 – 0.1 kg·m² for a small table, more for a loaded one. A solid disc has m·r²/2, a thin ring m·r². Step 2 can add it up from parts. ### Stroke How far the carriage travels in one move, from rest to rest. Unit: mm. Typical: 50 – 1 000 mm. The stroke and the move time set the speed and the acceleration; everything downstream follows from those two numbers and the mass. ### Angle How far the table turns in one move, from rest to rest. Unit: deg. Typical: 45 – 180° for an index, 360° for a full turn. ### Move time The time the move may take, from the first motion to standstill at the end. Unit: s. Typical: 0.2 – 2 s. Shorter is harder: halving the time doubles the peak speed and quadruples the acceleration — and the torque with it. ### Dwell How long the axis stands still at the end position before the next thing happens — clamping, drilling, waiting. Switch it on or off; off, the move back (or the next cycle) follows at once. Unit: s. Typical: 0.1 – 2 s; switched off for an axis that comes straight back. It costs no torque, but it is part of the cycle the motor heats up over: a longer dwell lowers the RMS torque and a stepper's duty cycle. ### Move back to the start The axis comes back to where it started, as part of the cycle: the same move reversed, or one with its own time. Switch it on and its torque, its speed, its braking energy and its bar are counted. Typical: on for an axis that returns every cycle — a pick-and-place, a feeder; off for an index that carries on. On the way back the friction and a nut's preload turn round with the motion; gravity and the external force keep their direction, so a lifted load is held back on the way down and that energy comes back to the drive. A pneumatic cylinder always comes back: its retract is its move back. ### Time back How long the move back takes. The same as the move out by default; a time of its own gets the same split and the same shape over that time. Unit: s. Typical: the same as the move out; longer for a gentle return, shorter when the empty trip may be quick. A faster move back sets the cycle's peak speed and acceleration — the motor's top speed and the screw's critical-speed check see it. ### Rest at home How long the axis stands at its start position after the move back, before the next cycle begins. Unit: s. Typical: 0 when the next move starts at once; the loading time of a pick-and-place. Like the dwell, it costs no torque and lengthens the cycle: a longer rest lowers the RMS torque, a stepper's duty cycle and the brake resistor's average power. It is the gap before the next bar on the cyclogram, not part of the bar. ### Profile shape How the speed rises and falls: with sharp corners (trapezoid) or rounded ones (S-curve). Typical: trapezoid for a first sizing; S-curve when the machine must run smoothly. The S-curve needs a higher peak acceleration for the same move time, but no sudden change of force — gentler on the structure, the belt and the workpiece. ### Ramp split What share of the move time goes to accelerating, cruising and decelerating. Unit: %. Typical: 33 / 33 / 33; 25 / 50 / 25 for a longer cruise. Equal thirds is the usual starting point. Shorter ramps mean a lower peak speed but a higher acceleration. ### Jerk time per ramp In an S-curve, the share of each ramp spent building the acceleration up or letting it fade out. Unit: %. Typical: 25 – 50 %. At 50 % the acceleration never holds — the smoothest shape, with the highest peak. Near 0 % it is a trapezoid again. ### Orientation of travel Whether the move runs flat, straight up, or up a slope — gravity works against the motor on anything but flat. Typical: horizontal for a table, vertical for a lift axis. A vertical or inclined axis needs torque even at rest, and a brake for when the power is off. ### Incline The angle of the travel above horizontal; the move goes up the slope. Unit: deg. Typical: 15 – 60°. Gravity pulls with sin(angle) of the weight along the slope; the friction works on cos(angle) of it. ### External force A force that pushes against the move: a spring, a cutting force, a cable chain, a seal. Unit: N. Typical: 0 for a free move; 50 – 500 N for a process force. Enter it positive when it opposes the move. It adds to the torque in every phase, cruise included. ### External torque A torque that resists the turn: a process load, a brush, bearing drag, a cable. Unit: N·m. Typical: 0 – 10 N·m. Positive when it opposes the move; it adds to the torque in every phase. ### Guide friction The friction coefficient of the guides the carriage runs on: the friction force is μ times the weight pressing on them. Typical: 0.005 – 0.02 for rolling guides, 0.1 – 0.2 for sliding ones. Rolling guides are nearly free; dovetails and plain bushings are not. Seals and wipers add a little either way. ### The motion chart Position, velocity, acceleration, jerk and motor speed over one cycle, each curve scaled to its own peak so they share one plot; the chips under it switch them on and off. Typical: position, velocity and acceleration on; jerk for an S-curve; the motor speed once the actuator is set. The shaded fields are the phases: blue while the axis accelerates, green while it cruises at constant speed, orange while it decelerates, grey for the dwell at the end position and the rest at home. The move back, when step 1 has one, is shaded the same way; its velocity runs below the line. Hover the chart to read the values at a moment; the cursor otherwise follows the drawing's clock. Zoom in on a part of it with the wheel while ctrl (⌘ on a Mac) is held, or with a pinch on a trackpad or a touch screen; drag to pan, double-click to see the whole again. The −, + and Whole buttons beside the chart do the same without a wheel. ## Ball screw, belt, rack and rotary table terms URL: https://cyclogram.app/calculations/glossary/actuators/ ### Actuator The mechanism that turns the motor's torque into the move — or, for a cylinder, the air that does it instead. Typical: ball screw for precision, belt for long fast strokes, rack for long travel, rotary table for indexing, cylinder for a two-position move. Each type has its own fields, and all five are computed to the same depth. A cylinder has no motor: step 4 checks its force and its stroke time instead. ### Screw lead How far the nut travels when the screw turns once. Unit: mm. Typical: 5 – 20 mm; 25 – 40 mm for fast axes. A longer lead gives more speed per motor turn and less thrust per unit of torque — and a lower screw speed for the same table speed. ### Screw diameter The nominal (outer) diameter of the screw shaft. Unit: mm. Typical: 12 – 25 mm for small axes, 32 – 50 mm for machine tables. It sets the screw's own inertia, its stiffness, its critical speed and the D·N value the nut's balls can take. ### Root diameter The diameter at the bottom of the thread groove — the shaft that actually bends. Unit: mm. Typical: about 0.8 – 0.9 × the nominal diameter. It is what the critical-speed formula uses. Leave it blank and the page takes 0.85 × the nominal diameter; the catalogue has the exact figure. ### Length between bearings The unsupported length of screw between its two bearing supports. Unit: mm. Typical: 300 – 1 500 mm. The critical speed falls with the square of this length: twice as long, a quarter of the speed. ### Screw efficiency How much of the motor's work reaches the carriage; the rest is friction in the nut. Typical: 0.9 for a ball screw, 0.3 – 0.5 for a trapezoidal thread. The torque for every force on the carriage is divided by it. ### Supports How the two ends of the screw are held: fixed (a pair of angular-contact bearings), supported (one bearing that lets the shaft tilt), or free. Typical: fixed – supported for most axes. Stiffer ends raise the critical speed: fixed – fixed takes 2.3 times the speed of supported – supported, and fixed – free barely a third of it. ### Nut preload The force a preloaded nut presses on the screw with, to take out the backlash. Unit: N. Typical: 0 for a plain nut; 5 – 10 % of the dynamic load rating for a preloaded one. Preload adds a constant drag torque — small, but there in every phase. ### Preload friction The friction coefficient the preload drag is computed with: drag torque = μ₀ × preload × lead / 2π. Typical: 0.1 – 0.3. Catalogues quote 0.1 to 0.3; 0.2 is a fair middle. ### Coupling inertia The inertia of the coupling between the motor shaft and the screw, the drive pulley, the pinion or the table. Unit: kg·m². Typical: 1 – 50 × 10⁻⁶ kg·m². It turns at motor speed, so it counts as it is — no gear ratio in between. Type it as 1e-6 for 1 × 10⁻⁶ kg·m². ### Screw inertia The inertia of the screw shaft about its own axis. Unit: kg·m². Typical: a solid steel Ø16 × 500 mm screw: 2.5 × 10⁻⁵ kg·m². Left blank, the page takes a solid steel cylinder of the given diameter and length, m·d²/8. The catalogue figure is usually a little lower — the thread is cut away. ### Pulley pitch diameter The diameter at which the belt's cords run — the pitch line — not the outer diameter of the teeth. Unit: mm. Typical: 25 – 80 mm; from the tooth count, D = z · pitch / π. Every force on the carriage reaches the motor through half of it and every mass through its square: a bigger pulley asks for more torque and reflects more inertia, but turns slower for the same speed. ### Belt mass The mass of the whole belt loop: it moves at the carriage's speed and counts as inertia with the carriage, (m + m_belt) · (D/2)². Unit: kg. Typical: a few hundred grams on a short axis; the catalogue gives the belt's mass per metre. It rides on no guide, so it adds no friction; its tension loads the bearings, not the motor. 0 if unknown — it is usually small next to the carriage. ### Pulley width The width of each pulley, used only to estimate its inertia as a solid cylinder of the pitch diameter. Unit: mm. Typical: the belt width plus the flanges: 20 – 50 mm. Type the pulleys' inertia from the catalogue instead when you have it; the estimate is then ignored. ### Pulley material What the pulleys are made of, for the inertia estimate: aluminium at 2 700 kg/m³ or steel at 7 850 kg/m³. Typical: aluminium for most axes; steel for small pulleys under high load. A steel pulley has 2.9 times the inertia of an aluminium one of the same size — on a direct-driven belt axis that is most of the mechanism's own inertia. ### Pulley inertia The inertia of both pulleys together, about their axes. Unit: kg·m². Typical: two solid aluminium Ø50 × 32 mm pulleys: 1.1 × 10⁻⁴ kg·m². Left blank, the page takes two solid cylinders of the pitch diameter and width in the chosen material, m·D²/8 each. The catalogue figure counts the bore and the flanges and is a little different. ### Belt efficiency How much of the motor's work reaches the carriage; the rest is lost in the belt's flexing and the pulleys' bearings. Typical: 0.9 – 0.98 for a toothed belt; 0.9 is the usual sizing value. The torque for every force on the carriage is divided by it; while braking, the load drives the motor through it instead. ### Allowable belt force The largest peripheral force the belt may carry, as its catalogue states it for the width, the pitch and the teeth in mesh. Unit: N. Typical: from the belt catalogue, for the width and the pitch; blank = no check. Step 3 compares it with the force the belt carries at the peak of the move — friction, gravity and the external force plus (m + m_belt) · a — and flags the belt when it is over. Leave it blank and no check is made. ### Module The size of the teeth: the pinion's pitch diameter divided by its number of teeth. Pinion and rack must have the same module. Unit: mm. Typical: 1 – 6 mm; 2 or 3 for most machine axes. A bigger module carries more force at the teeth, but for the same tooth count it makes a bigger pinion — more torque at the motor for every newton at the carriage. Along the rack, tooth to tooth, the pitch is π · m. ### Pinion teeth How many teeth the pinion has. With the module it sets the pitch diameter: D = m · z. Typical: 17 – 30; fewer than 17 needs a profile-shifted pinion. Fewer teeth make a smaller pinion: less torque and less reflected inertia at the motor, but more turns for the same travel — a faster motor. Below about 17 teeth a standard 20° pinion undercuts its own teeth; the calculator takes 8 or more. ### Pinion face width The width of the pinion's teeth along its axis, usually the width of the rack. Unit: mm. Typical: 8 – 12 × the module; 20 mm for module 2. The calculator uses it to estimate the pinion's inertia, as a solid steel disc of the pitch diameter — and, with the motor fixed, the moving rack's mass. Type the pinion's inertia instead and the width is not needed. ### Pinion inertia The pinion's moment of inertia about its own axis, from its catalogue. Unit: kg·m². Typical: blank = a solid steel disc of the pitch diameter and the face width. Left blank, it is estimated: m = ρ · π · D² / 4 · w and J = m · D² / 8 — a fair figure for sizing; a pinion with a large bore is a little lighter. It turns at the pinion's speed, so the gearbox divides it by the square of the ratio. ### Rack-and-pinion efficiency How much of the motor's work reaches the carriage through the mesh; the rest is lost in the teeth and the pinion's bearings. Typical: 0.9 – 0.98; 0.9 is the usual sizing value. The torque for every force on the carriage is divided by it; while braking, the load drives the motor through it instead. ### Allowable force at the teeth The largest tangential force the teeth may carry, as the rack's catalogue states it. Unit: N. Typical: from the rack's catalogue, for the module, the width and the quality; blank = no check. Step 3 compares it with the force at the teeth at the peak of the move — friction, gravity and the external force plus the moved mass times its acceleration — and flags the rack within 10 % of it or over it. Leave it blank and no check is made. ### Where the motor sits On the carriage, the rack is bolted to the frame and the pinion runs along it, the motor and gearbox riding with the load. On the frame, the motor and pinion stay put and push the rack, which moves with the load. Typical: on the carriage for long axes and shared racks; on the frame for shorter strokes. The formulas are the same either way; what moves is not. The page adds the motor's mass in the first case and the rack's in the second — to the inertia, the friction and, on a climbing axis, the weight the motor lifts. ### Riding with the pinion The mass that rides on the carriage with the pinion when the motor sits there: the motor, the gearbox, their bracket — and the cable chain's share, if you count it. Unit: kg. Typical: 1 – 5 kg for a servo, its gearbox and the bracket. It moves with the load at the load's speed, so it is added to the moving mass: more torque to accelerate, more friction, more weight on a climbing axis. 0 if nothing rides along. ### Rack length The length of the rack when the motor is fixed and the rack moves with the load. Unit: mm. Typical: blank = the stroke + 200 mm; at least the stroke plus the pinion's diameter. It sets the rack's mass when that is left blank. Shorter than the stroke plus the pinion's mesh, and the pinion runs off the end — step 3 says so. ### Rack mass The mass of the moving rack, best from its catalogue: the mass per metre times the length. Unit: kg. Typical: blank = 7850 · w² · L, a square steel bar of the face width. Left blank, it is estimated as a solid square steel bar as wide as the face width — a little more than a real rack, whose tooth gaps take some steel away. It moves with the load, so it is added to the moving mass. ### Table friction torque The torque the table's bearings and seals need just to turn, measured at the table. Unit: N·m. Typical: 0 – 0.5 N·m for a table on good bearings; from its catalogue. It acts against the turn the whole way, like the external torque of step 1, and reaches the motor divided by the gear ratio and the efficiencies. 0 when nothing rubs. ### Table drive efficiency How much of the torque into the table reaches the load, for a table with its own drive inside — a worm, a belt or a cam. Typical: 1 for a table on the motor or the gearbox; 0.5 – 0.9 for a worm or a belt inside it. A table that sits straight on the motor or the gearbox has nothing in between: leave it at 1 and the gearbox's own efficiency counts. A table with a reduction inside takes its ratio in the gearbox fields. ### The load, part by part Everything that moves with the axis, one line each, so the total stays traceable. Typical: carriage, workpiece, fixture, nut. On a linear axis the masses add up. On a rotary axis each part is an inertia, or a disc given by mass and diameter (m·d²/8). ### Gearbox A reducer between the motor and the screw, the drive pulley, the pinion or the table: the motor turns faster and sees less inertia. Typical: none for a direct-coupled screw; 3 – 10 : 1 for a belt axis or where the motor is small. The load inertia at the motor falls with the ratio squared — the usual cure for an inertia ratio that is too high. ### Gear ratio Motor turns per turn of the screw, the drive pulley, the pinion or the table — 3 means the motor turns three times while it turns once. Typical: 3 – 10. Speed at the motor rises by the ratio, torque at the motor falls by it, inertia at the motor falls by its square. ### Gearbox efficiency How much of the motor's torque comes out of the gearbox. Typical: 0.9 – 0.97 for a planetary stage. ### Gearbox inertia The gearbox's own inertia, as the catalogue states it at the input (motor) shaft. Unit: kg·m². Typical: 1 – 100 × 10⁻⁶ kg·m². Type it as 1e-5 for 1 × 10⁻⁵ kg·m². ## Pneumatic cylinder terms URL: https://cyclogram.app/calculations/glossary/cylinders/ ### Bore The inside diameter of the barrel — the piston's diameter. The force out is the pressure on the piston's area: F = p · π · d² / 4, of which the seals leave about 88 %. Unit: mm. Typical: 12 – 100 mm; ISO 15552 sizes 32, 40, 50, 63, 80, 100. A bigger bore pushes harder but fills with more air each stroke, so on the same flow it strokes more slowly. The right bore is the smallest whose load ratio stays under 0.5. ### Rod diameter The diameter of the piston rod. On the retract the air pushes only on the ring around it, so the force in is smaller than the force out. Unit: mm. Typical: blank = the usual ISO rod: 12 mm for a 32 bore, 16 for 40, 20 for 50 and 63. Leave it blank and the calculator takes the usual rod of an ISO cylinder of that bore; type it for any other cylinder. ### Supply pressure The gauge pressure at the valve — what the regulator's gauge reads, above the atmosphere around it. Unit: bar. Typical: 6 bar is usual; 4 – 8 bar in most factories. The force rises with it, and so does the air each stroke takes: the barrel's volume times (p + 1.013) / 1.013 in litres of free air. ### Air flow to the cylinder How much air reaches the cylinder per minute, in litres of free air at atmospheric pressure — the least of what the valve, the fittings and the tube let through. Unit: l/min. Typical: 100 – 1 000 l/min of free air, from the valve's rated flow. The stroke time is mostly the air the stroke needs divided by this flow. A valve's nominal flow, measured at 6 bar with a 1 bar drop, is an upper bound; long thin tubes and small fittings lower it, and the calculator does not model them — fold them into this figure. ### Tube length The length of the tube from the valve to the cylinder's port. Unit: m. Typical: 0.5 – 3 m; 0 for a valve on the cylinder. The tube fills and empties with every stroke, so its volume adds to the air each stroke needs and to the time it takes. Its flow resistance is not modelled — a long tube also lowers the air flow. ### Tube inside diameter The inside diameter of the tube, not the size it is sold by — tubes are named by their outside. Unit: mm. Typical: 4 mm inside a 6 mm tube, about 6 inside an 8, about 7.5 inside a 10. With the length it gives the tube's volume, π · d² / 4 · L, filled with every stroke. ### Flow control The share of the air flow the flow-control valve lets through: 1 when it is open, less as it is turned in. Typical: 1 = fully open; 0.3 – 0.8 when a throttle slows the stroke. Throttling the exhaust is how a cylinder is slowed and smoothed. The flow time is divided by it, so 0.5 doubles it. ### Valve delay The time between the signal and the piston starting to move: the valve switching, and the pressure building until the piston breaks loose. Unit: s. Typical: 0.02 – 0.1 s with the control's own response. It is added to the stroke time and drawn as the pale band at the start of the chart in step 3. ### Cushioning The extra time the end cushion takes: the last millimetres of the stroke are slowed so the piston does not strike the end cap. Unit: s. Typical: 0.02 – 0.1 s; 0 without end cushions. Adjustable cushions trade time for a softer stop. It is added to the stroke time and drawn as the slowing at the end of the stroke. ### Load ratio The force the load needs divided by the force the cylinder makes at this pressure, after the seals' share. Typical: under 0.5 for a quick stroke; 0.5 – 0.8 slows it; over 0.8 it can stall. Under 0.5 the air, not the load, sets the speed. Above it the stroke slows — the calculator stretches the flow time by 1 / (1 − ratio), an empirical model, indicative above 0.5 — and above 0.8 it crawls and can stall. At 1 or more the piston does not move at all. ## Motor sizing terms URL: https://cyclogram.app/calculations/glossary/motors/ ### Motor The motor that drives the axis: pick one from the list, or type a catalogue line in by hand and keep it under "my motors". Typical: a generic 3000 rpm servo of 50 W to 3 kW to start; your own motors, saved in this browser; the catalogues later. The checks below the list say whether it makes the move, and with how much to spare. ### Motor type What sort of motor it is — the type decides what its numbers mean, the shape of its torque over speed, and which checks apply. Typical: a servo for positioning; a stepper for small, cheap, open-loop axes; a BLDC motor for speed control; an AC induction motor on a VFD for big, slow axes. A servo gives its rated torque to the rated speed and its peak for a short while, at constant power above. A stepper holds its pull-out torque to a corner speed and loses it fast beyond; the page uses that torque over a safety factor and watches the duty cycle. A BLDC motor keeps its rated torque over the whole speed range, with a limited-duty torque for a few seconds. An AC induction motor on a drive gives its rated torque to the base speed and weakens beyond, its breakdown torque falling with the square of the speed. ### Rated torque The torque the motor can give continuously without overheating — the line the RMS torque is checked against. Unit: N·m. Typical: 0.3 N·m (100 W) to 10 N·m (3 kW) at 3000 rpm. For a servo it is the rated power divided by the rated angular speed: P / (2π·n/60). ### Peak torque The torque the motor can give for a short while — the line the peak of the move is checked against. Unit: N·m. Typical: about three times the rated torque. Above the rated speed it falls off; the page derates it at constant power unless the motor carries its own curve. ### Rated speed The speed up to which the motor gives its full torque. Unit: rpm. Typical: 3000 rpm for most servos; 1500 or 2000 for big ones. Beyond it the available torque drops, roughly as rated speed over actual speed. ### Maximum speed The speed the motor must never exceed — mechanically and for its feedback. Unit: rpm. Typical: 5000 – 6000 rpm. The move's top motor speed is checked against it. ### Rotor inertia The inertia of the motor's own rotor, as the catalogue states it. Unit: kg·m². Typical: 3 × 10⁻⁶ (50 W) to 6 × 10⁻⁴ (3 kW). It sets the inertia ratio, load over rotor, which decides how well the servo loop can be tuned. Type it as 2.6e-5 for 2.6 × 10⁻⁵. ### Voltage The supply the motor and its drive are built for. Unit: V. Typical: 230 V single-phase up to about 750 W; 400 V three-phase above. It does not enter the sizing; it is kept so the saved motor is complete. ### Brake Whether the motor has a holding brake — a spring-applied brake that holds the shaft when the power is off. Typical: on vertical axes; off elsewhere. Needed on a vertical axis, so the load does not fall when the drive is disabled. With a brake, the dwell can be held by it instead of by the motor's torque. ### Hold with the brake During the dwell the brake holds the load, so the motor carries no torque then. Typical: on, when the brake may be engaged during the dwell. That lowers the RMS torque on a vertical axis. Engaging the brake every cycle wears it; check the brake's cycle rating. ### Inertia ratio limit The largest load-over-rotor inertia ratio you accept. Typical: 10 for standard servos; 5 for very dynamic axes; up to 30 with a stiff coupling and good tuning. Below it the servo loop settles quickly; above it the motor can only tune slowly or oscillates. ### Holding torque The torque a stepper holds at standstill with its rated current — the figure on the catalogue's first line. Unit: N·m. Typical: 0.4 N·m (NEMA 17) to 8.5 N·m (NEMA 34). Its pull-out torque, what it gives while turning, starts near the holding torque and falls with speed; the page takes it flat to the corner speed and falling as 1/n beyond unless a curve is given, and uses it over the safety factor. ### Speed the torque holds to The speed up to which the stepper's pull-out torque stays near its holding torque; beyond it the torque falls away, roughly as that speed over the actual speed. Unit: rpm. Typical: 200 – 600 rpm on a chopper driver; higher with more supply voltage. A catalogue's pull-out curve says exactly how; without one the page takes the corner from this field. ### Safety factor How much of the pull-out torque a stepper is allowed to use: the move's torque times this factor must stay under the pull-out curve. Typical: 2; 1.5 for a well-known load. Open loop forgives nothing — a stall is a lost position — so steppers are sized with twice the torque they need. ### Step angle The angle of one full step; with the speed it gives the step rate the driver must deliver. Unit: deg. Typical: 1.8° (200 steps per turn); 0.9° for fine steppers. It does not change the torque the page computes — it is shown with the pull-out check as steps per second, the unit catalogue curves are drawn over. ### Limited-duty torque The torque a BLDC motor's controller allows for a short while — starting, accelerating — before it limits the current. Unit: N·m. Typical: 1.5 – 2 × the rated torque, for a few seconds. The peak of the move is checked against it; the RMS against the rated torque. ### Permissible load inertia The largest load inertia the maker allows at the motor shaft for the controller to hold its speed through the ramps. Unit: kg·m². Typical: from the catalogue; blank = the ratio rule. Given, the page checks the reflected load inertia against it; blank, it applies the inertia-ratio limit instead. ### Breakdown torque The most an induction motor can give before it stalls — the top of its torque–speed curve. Unit: N·m. Typical: 2 – 3 × the rated torque. The peak of the move is checked against it; above the base speed it falls with the square of the speed, and the drive's own current limit may sit below it. ### Base speed The speed the motor reaches at its rated frequency, just under the synchronous speed; up to it the drive gives full torque. Unit: rpm. Typical: 1 400 – 1 450 rpm for a 4-pole motor at 50 Hz (1 750 at 60 Hz); 2 900 for 2-pole. Above it the drive weakens the field: constant power, the torque falling as base speed over actual speed. ### Catalogue curves The motor's real speed–torque outline, typed in from the datasheet: one point per line, the speed in rpm, then the torque in N·m. Unit: rpm, N·m. Typical: 5 – 15 points read off the datasheet's speed–torque chart, at your drive voltage. Without them the page draws a generic outline — torque flat to the rated or corner speed, falling as constant power beyond — a stand-in, not the motor. With them the map draws the sheet's line through your points and every check reads the torque at speed from it: a servo's peak and continuous curves, a BLDC motor's limited-duty and rated, an induction motor's breakdown and rated, a stepper's pull-out (its continuous line is the pull-out over the safety factor). Read the curve for the drive voltage you will use; a stepper's pull-out curve moves a long way with the supply. Give it to the motor's maximum speed: beyond your last point the page assumes no torque at all. Part the two numbers with a comma, a space, a tab, a colon or an arrow; write decimal commas only with a space or a tab between the numbers. A line that cannot be read is named under the box and the points in force stay until it is fixed. The same points travel in motor files as curveCont and curveInt, as a list of pairs or as this text. ### Motor files Your own motors as files: drop them here and they join "My motors" in this browser; download the list to keep or share it. Typical: one JSON file per motor, or a CSV with one motor per row; a whole folder at once. A JSON file holds one motor or a list, with the page's field names (ratedTorque_Nm, peakTorque_Nm, ratedSpeed_rpm, maxSpeed_rpm, rotorInertia_kgm2, voltage_V, kind, name, maker …); a CSV has those as column headers, with common spellings understood. A folder is read through, file by file; one with the same name and maker replaces the saved one. ### Compare motors Up to two more motors judged on the same move, each with its own colour on one speed–torque map and in one table — each by the rules of its own kind. Typical: up to two more motors beside the one chosen — from the lists or your own. Each motor's two outlines are drawn in its colour: solid, the torque it gives for a short while; dashed, the torque it holds. The black line is what the move asks, drawn once, through the chosen motor; through the others it would differ only by their rotor inertia, and their own peaks stand in the table. A servo is held to its intermittent envelope at the peak and to its rated torque for the RMS; a stepper to its pull-out torque over the safety factor, with a duty cycle under 50 % instead of an RMS; a BLDC motor to its limited-duty torque; an AC induction motor to its breakdown torque, which falls with the square of the speed above the base speed. Step 5 lists the regeneration of every motor compared; the formulas and the torque by phase below the table are the chosen motor's. ### The speed–torque map What the motor can give at each speed, and what the move asks of it at each speed — drawn below the motor, with the actuator optionally running underneath on the same clock as the ball. Typical: the black line inside the solid outline with its peak marked; the dotted RMS line inside the dashed one. The motor's two outlines are in its colour. The solid one is the torque it gives for a short while: a servo's peak, a stepper's pull-out, a BLDC motor's limited-duty, an induction motor's breakdown torque. The dashed one is the torque it holds without overheating: the continuous or rated torque, for a stepper the pull-out torque over the safety factor. Both fall off above the rated speed. The black line is the move itself, torque against speed through one cycle, its peak marked by a dot; the dotted line is the RMS over the cycle; the blue ball is the moment the drawing is at. "What is what" writes these meanings onto the chart; the switch under it shows the actuator moving with the ball. Zoom in on a part of it with the wheel while ctrl (⌘ on a Mac) is held, or with a pinch on a trackpad or a touch screen; drag to pan, double-click to see the whole again. The −, + and Whole buttons beside the chart do the same without a wheel. ### Torque over the cycle The torque the motor must give at every moment of the cycle. Typical: a step up while accelerating, a small constant torque while cruising, a step down below zero while braking. The shaded fields are the phases: accelerating (blue), cruising (green), decelerating (orange), dwelling and resting (grey); the move back, when there is one, is shaded the same way with its torque below the line. Where the black curve dips below zero the load drives the motor — it brakes — and that is the energy step 5 counts. The dashed line in the motor's colour is the torque available at that moment's speed; the dotted black line the RMS over the cycle, which the rated torque must cover. Motors compared add their own dashed lines. "What is what" writes these meanings onto the chart. Zoom in on a part of it with the wheel while ctrl (⌘ on a Mac) is held, or with a pinch on a trackpad or a touch screen; drag to pan, double-click to see the whole again. The −, + and Whole buttons beside the chart do the same without a wheel. ## Regeneration and cyclogram terms URL: https://cyclogram.app/calculations/glossary/regeneration/ ### Energy the drive can absorb How much braking energy the drive's DC bus capacitors can take in before the voltage reaches the brake chopper's threshold. Unit: J. Typical: 10 – 30 J for a small drive, 50 – 200 J for a large one. The drive's data sheet gives it, or work it out from the capacitance and the two voltages below. ### Bus capacitance The capacitance of the drive's DC bus. Unit: µF. Typical: 500 – 2 000 µF for a small drive. With the bus voltage and the chopper threshold it gives the energy the bus can take: ½·C·(V_max² − V_bus²). ### Bus voltage The DC bus voltage while nothing is braking — the mains voltage rectified, √2 times its RMS value. Unit: V. Typical: about 320 V on a 230 V supply, 565 V on 400 V. ### Chopper threshold The bus voltage at which the drive switches the brake resistor in. Unit: V. Typical: about 390 V on a 230 V supply, 760 V on 400 V. The capacitors absorb energy between the bus voltage and this threshold; beyond it the resistor must burn the rest. ### Regeneration efficiency The share of the braking work that reaches the DC bus; the motor's copper losses and the drive's losses take the rest. Typical: 0.8 – 0.95. ### Row name The name of the lane the move lands on in your cyclogram. Typical: what the actuator is called on the machine: "Feeder X", "Lift Z". Blank takes the calculation's name.