Servo Motor Sizing Calculator (Inertia Ratio and RMS Torque)

How the calculator works

The tool takes servo sizing apart into two questions and answers them in order. First, can the drive tune the axis? That is the inertia question, and it comes down to the reflected load inertia against the rotor inertia. Second, will the motor overheat or stall? That is the torque question, and it comes down to the RMS torque and the peak torque across the move. You describe the mechanism, the load, the ratio, the rotor, and the move profile, and the panel returns the reflected inertia, the inertia ratio with a pass or fail flag, the angular acceleration, the acceleration torque, the peak torque, and the RMS torque, then draws the torque across accel, run, decel, and dwell so you can see the shape of the demand.

The first thing it settles is the reflected load inertia, because both answers depend on it. A servo feels the load as an inertia at the motor shaft, and that depends on the mechanism. A ball screw turns the linear mass of the moved load into a rotary inertia through the lead, then adds the inertia of the screw body itself. A belt and pulley turns the mass into an inertia through the pulley radius. A known load inertia comes straight in. Whatever the mechanism, a gear reducer between the load and the motor divides that reflected inertia by the square of the ratio, which is the single most useful lever in the whole calculation.

From the reflected inertia the inertia ratio falls straight out. The ratio is the reflected load inertia divided by the motor rotor inertia, and it is the number that tells you whether the axis will be tunable. On the default, the reflected inertia is 1.01 kg.cm2 and the rotor is 0.20 kg.cm2, so the ratio is 5.0:1, which the tool flags as within range. Keep the ratio at or below 10:1 for general motion, and nearer 3:1 to 5:1 for a high-response axis, and the drive can hold the axis steady. Push it far past 10:1 and the load starts to dominate the motor, the axis overshoots, and no amount of gain will settle it cleanly.

The torque side needs the move. The tool builds a trapezoidal profile, a ramp up to speed, a run at constant speed, a ramp down, and a dwell at rest, from the speed and the four times you enter. During the ramp up it needs enough torque to accelerate the combined inertia of the rotor and the reflected load, plus the constant load torque. That acceleration torque is the biggest instantaneous demand, and adding the load torque gives the peak. During the run it only has to hold the load, during decel the acceleration term flips sign and helps, and during the dwell it does nothing. The tool then takes the heating-equivalent RMS torque across the whole cycle, which is the number the motor’s continuous rating must clear. On the default the peak is 0.48 N.m and the RMS is 0.18 N.m. The reflected inertia and the ratio also feed the wider machine-sizing work in the Industrial Automation silo, where the reducer that fixes a high ratio is sized with the Gear Ratio and Gearmotor Torque Calculator.

Reflected load inertia by mechanism

The reflected inertia is the inertia of the load as the motor feels it at its own shaft, and it depends entirely on how the load is coupled. This matters because a servo does not act on the load directly. It acts on the rotor, and the mechanism between the rotor and the load transforms the load’s mass or inertia into an equivalent inertia at the motor. Get that transform right and the rest of the sizing follows. Get it wrong and every torque and ratio downstream is off. The tool handles the three mechanisms that cover most industrial axes: the ball screw, the belt and pulley, and a directly coupled or known load inertia through a reducer.

A ball screw turns rotation into linear motion, so the moved mass reflects as a rotary inertia through the lead. The reflected inertia of the mass is the mass times the lead over two pi, squared: J = m x (lead / (2 pi))^2. The lead is the linear distance the load travels per turn of the screw, so a fine lead couples the mass loosely and reflects a small inertia, while a coarse lead couples it tightly and reflects a large one. The screw body has its own inertia too, treated as a solid cylinder, one half the screw mass times the screw radius squared: J = 1/2 x m_screw x r_screw^2. You add the two, then divide by the square of the gear ratio if a reducer sits in the drive. On the default, 20 kg on a 10 mm lead reflects as a small inertia, and the 1 kg screw of 20 mm diameter adds its own, summing to 1.01 kg.cm2 at a ratio of 1.

A belt and pulley, or a rack and pinion, couples the mass through the pulley radius rather than a lead. The reflected inertia of the mass is the mass times the pulley radius squared: J = m x r_pulley^2, then divided by the square of the ratio. A bigger pulley moves the load faster for the same motor speed but reflects a larger inertia, so the pulley size trades speed against the inertia the motor feels, the same trade the lead makes on a screw. A directly coupled load, or one whose inertia you already know from a CAD model or a catalogue, comes in as a known inertia and only needs the reducer term: J_reflected = J_load / i^2. A rotary index table, a drum, or a flywheel is usually sized this way, because the load inertia is easier to read off the part than to build from a mass and a radius.

The inertia ratio and the 10:1 rule

The inertia ratio is the reflected load inertia divided by the motor rotor inertia, and it is the first test a servo has to pass. It measures how much bigger the load looks to the motor than the motor itself. A ratio of 1:1 means the load and the rotor are matched, which is the ideal a servo designer aims for on the sharpest axes. A ratio of 5:1 means the load looks five times heavier than the rotor, which is comfortable for general motion. A ratio of 40:1 means the load swamps the rotor, and the drive will fight to control it. The tool computes the ratio and flags it against the usual limits, so on the default a 5.0:1 ratio comes back marked good.

The rule of thumb the whole industry uses is to keep the ratio at or below 10:1 for general motion, and between 3:1 and 5:1 for a high-response or precise axis. The reason is control. A servo loop reads the motor’s position and corrects it thousands of times a second, but the load is connected to the motor through a coupling, a screw, or a belt that has some flex in it. When the load inertia is close to the rotor, the motor can start and stop it crisply, and the loop stays stable at high gain. When the load inertia dwarfs the rotor, the flexible coupling behaves like a spring between a small mass and a large one, the axis rings, and every attempt to raise the gain to sharpen the response makes the overshoot worse instead of better.

Above 10:1 you are into the territory where the axis is hard to tune and prone to overshoot, and the fix is almost always a reducer rather than a bigger motor. A reducer cuts the reflected inertia by the square of the ratio, so it is a far more powerful lever than picking a motor with a slightly larger rotor. This is the point where the Gear Ratio and Gearmotor Torque Calculator becomes the natural next step: it sizes the reducer that brings a runaway inertia ratio back into range. A 3:1 reducer, for instance, cuts the reflected inertia to about a ninth, which turns a 40:1 axis into a tunable one, as the fourth worked example below shows. The trade is that the reducer also multiplies the torque the motor must supply and cuts the top speed, so you look for the ratio that lands the inertia in range without asking for more torque than the motor has.

The trapezoidal move and acceleration torque

The torque a servo needs is set by the move, not just the load, so the tool builds the move as a trapezoidal profile and works the torque through each segment. A trapezoidal move has four parts: the motor ramps the speed up from rest to the move speed over the accel time, holds the move speed over the run time, ramps back down to rest over the decel time, and sits still through the dwell time. Plotting the speed against time draws a trapezoid, which is where the name comes from. It is the simplest realistic profile, and it is what most point-to-point axes actually run, so it is the right shape to size against.

The demanding part is the acceleration. To ramp the speed up, the motor has to overcome the inertia of everything it is spinning, the rotor plus the reflected load, at the rate the ramp demands. That rate is the angular acceleration, alpha, which is the speed change in rad/s divided by the accel time. The tool works it from the rpm: alpha = 2 pi x delta_n / (60 x t_accel), where delta_n is the speed change in rpm. On the default, going to 3000 rpm in 0.1 s gives an angular acceleration of 3,142 rad/s2, a brisk ramp. A shorter accel time raises alpha in direct proportion, which is why a fast move is so much more demanding than a slow one carrying the same load.

The acceleration torque is that angular acceleration times the combined inertia: T_a = (J_motor + J_reflected) x alpha. It is the torque that goes purely into speeding the mass up, and it is the largest single term in the whole move. On the default, the combined inertia of 0.20 plus 1.01 kg.cm2 accelerated at 3,142 rad/s2 gives an acceleration torque of 0.38 N.m. The full torque in each segment then adds the constant load torque on top. During the accel the torque is T_a plus the load, which gives the peak of 0.48 N.m. During the run it is just the load. During the decel the acceleration term reverses sign, because the motor is now braking the mass, so the torque is minus T_a plus the load. During the dwell the torque is zero. That sign flip on decel is why the braking phase is usually the gentlest part of the move, since the inertia is helping rather than fighting.

Peak torque versus RMS torque

A servo is rated two ways, and the move produces two torque numbers to match them. The peak torque is the largest instantaneous torque anywhere in the move, which is the acceleration torque plus the load, and it happens during the ramp up. On the default that is 0.48 N.m. The motor’s peak torque rating, the short-burst torque it can produce for a fraction of a second, must clear this number, or the axis will not accelerate as fast as the profile asks and it may stall on the ramp. Peak torque is a stall-and-follow test: can the motor make the instantaneous torque the fastest part of the move demands?

The RMS torque is a different animal. It is the heating-equivalent torque, the constant torque that would heat the motor as much as the real varying torque does across the whole cycle. Because heating goes with the square of the current, and current goes with torque, you square the torque in each segment, weight it by the segment time, sum those over the cycle, divide by the total cycle time including the dwell, and take the square root: RMS = square root of ( sum of torque^2 x time over the segments, divided by the total cycle time ). On the default that comes to 0.18 N.m. The motor’s continuous, or rated, torque must clear this number, or the motor will overheat over a long run of cycles even though it never stalls.

The reason a servo needs both tests is that the two can pull apart. A short, hard move with a long dwell can have a high peak and a low RMS: the motor works hard for a moment, then rests long enough that the average heating stays modest. A long, steady move with no dwell can have a modest peak and a high RMS: the motor never spikes, but it never rests either, so the heat builds. A motor can pass one test and fail the other. This is why you never size a servo on a single torque figure. You check the peak against the peak rating and the RMS against the continuous rating, both against the maker’s torque-speed curve, because the curve shows how both ratings fall off as the speed climbs. The dwell is the lever that separates the two, since a longer dwell lowers the RMS without touching the peak.

The selection rule

Once the tool has the reflected inertia, the inertia ratio, the peak torque, and the RMS torque, the selection rule is a clean pass or fail with three parts. The chosen servo must have a continuous, or rated, torque at or above the RMS torque, so it does not overheat. It must have a peak torque at or above the maximum instantaneous torque in the move, so it does not stall on the ramp. And it must give an inertia ratio inside the range, at or below 10:1 for general motion and nearer 3:1 to 5:1 for a high-response axis, so the drive can tune it. A servo that clears all three is the right size. A servo that misses any one of them is the wrong size, even if it clears the other two comfortably.

On the default, the rule reads straight off the panel. The RMS is 0.18 N.m, so the servo’s rated torque has to clear 0.18 N.m. The peak is 0.48 N.m, so its peak rating has to clear 0.48 N.m. The ratio is 5.0:1, which is inside the 10:1 limit and comfortable even for a fairly responsive axis. A small servo with a rated torque around 0.3 to 0.5 N.m and a peak of 1 N.m or more would clear all three with margin, so the default axis is easy to serve. The tool states the pass condition in its note so you can carry the two torque numbers straight to a catalogue and read down the ratings.

The one subtlety in the rule is the speed. Both the rated torque and the peak torque of a servo fall off as the speed rises, because the drive voltage runs out of headroom to push current into the motor at high rpm. So the numbers you compare against are not the flat catalogue figures at zero speed; they are the torque the motor can make at the speed the move actually runs at. The tool gives you the torque the move demands, and you read the maker’s torque-speed curve at your move speed to get the torque the motor can supply there. On a fast axis this matters, because a motor that clears the peak at low speed can fall short at 4000 or 5000 rpm where the curve has drooped.

Using a reducer to fix the ratio

When the inertia ratio comes out too high, the reducer is the fix, and it is a powerful one because the ratio enters as a square. The reflected inertia through a reducer is the load inertia divided by the ratio squared, so a 3:1 reducer cuts the reflected inertia to a ninth, a 5:1 reducer to a twenty-fifth, and a 10:1 reducer to a hundredth. That squared term is why a reducer beats a bigger motor almost every time for an inertia problem. Doubling the rotor inertia only halves the ratio, and it costs you a heavier, more expensive motor. A modest reducer can drop the ratio by a factor of ten or more and often costs less.

The fourth worked example is exactly this case. A 20 kg.cm2 load direct on a 0.5 kg.cm2 rotor gives a 40:1 ratio, far too high, and the axis will oscillate no matter how the drive is tuned. Add a 3:1 reducer and the reflected inertia falls from 20.00 to 2.22 kg.cm2, about a ninth, so the ratio drops to 4.4:1, which is comfortably in range. The acceleration torque at the motor also falls, from 6.44 to 0.86 N.m, and the RMS from 3.22 to 0.43 N.m, because the motor is now accelerating a much smaller reflected inertia. The Gear Ratio and Gearmotor Torque Calculator is the tool for picking that reducer, since it sizes the ratio, the output torque, and the gearbox rating the reduction needs.

The reducer is not free, and the trade is worth stating plainly. A reducer multiplies the torque the motor must supply by roughly the ratio, and it divides the top speed the load can reach by the ratio, so you cannot fix an inertia problem with an arbitrarily large ratio. Too small a ratio and the reflected inertia still swamps the motor; too large and the load can no longer reach the speed it needs, or the torque demand climbs past the motor’s rating. The sweet spot is the ratio that lands the inertia in range while leaving the top speed and the torque within what the motor can do. On many servo axes that lands between 3:1 and 10:1, which is why servo gearheads cluster in that range. A planetary gearhead, common on servo drives, packs that ratio into a short housing with low backlash, which matters because backlash in the reducer eats into the precision the servo is there to deliver.

Units and conventions

Servo sizing mixes units from the linear and the rotary worlds, so the tool keeps a consistent set and converts internally. Inertia is entered and reported in kilogram centimetre squared, kg.cm2, because that is the unit servo makers print on their datasheets, but the physics runs in kilogram metre squared, kg.m2, and one kg.cm2 is 1e-4 kg.m2. The tool shows both, so the reflected inertia of the default reads 1.01 kg.cm2 and 0.101 e-3 kg.m2 side by side. Torque is in newton metres, N.m, throughout, since that is the standard for servo ratings worldwide. Mass is in kilograms, length in millimetres for the lead and the diameters, speed in rpm, and time in seconds.

The angular quantities carry their own units. Angular acceleration is in radians per second squared, rad/s2, which is why the speed in rpm has to be converted through the 2 pi over 60 factor before it enters the acceleration formula. On the default, 3000 rpm becomes an angular acceleration of 3,142 rad/s2 over the 0.1 s ramp. The tool handles these conversions so you enter the numbers in the units your datasheets use and read the results in the same units, without carrying the rotation factors by hand. Where a diameter is entered, such as the screw diameter or the pulley diameter, the tool halves it to a radius internally, since the inertia formulas take the radius.

Where this calculator fits

It suits anyone specifying a servo axis without opening a full motion-control study. A machine builder laying out a new pick-and-place, a gantry, or an index table can fix the mechanism and the move, read the inertia ratio and the two torque numbers, and pick a servo and a gearhead straight from a catalogue. A controls engineer commissioning an axis that will not tune can check whether the inertia ratio is the culprit and size the reducer that fixes it. A designer trading off the lead of a screw or the size of a pulley can watch the reflected inertia and the ratio move as the geometry changes, and see where the axis crosses from tunable to twitchy.

Because it separates the inertia question from the torque question, it also builds the intuition that servo sizing runs on. You can watch the reflected inertia collapse when you add a reducer, see the peak torque climb as you shorten the accel time, and watch the RMS fall as you lengthen the dwell. This is the finale of the Industrial Automation silo, and it sits alongside the tools that size the rest of the drive. The Gear Ratio and Gearmotor Torque Calculator sizes the reducer that fixes a high inertia ratio, the Conveyor Belt Speed and Motor Power Calculator turns a shaft speed into a line speed, the Pneumatic Cylinder Force and Air Consumption Calculator covers the air-powered actuators alongside a servo axis, the Air Receiver Tank Size and Compressed-Air Demand Calculator sizes the compressed-air storage behind a shop, and the VFD Energy Savings Calculator shows what slowing a pump or fan saves. For the energy the whole line uses, the Energy Management hub rolls the running cost into the plant-wide picture, and Lean Production carries the tools for the flow around the machine.

Five worked examples

Example 1: a ball-screw axis (the default)

This is the case the tool opens on. A ball screw moves 20 kg on a 10 mm lead, with a 1 kg screw of 20 mm diameter, at a gear ratio of 1, driven by a motor with a 0.20 kg.cm2 rotor. The move goes to 3000 rpm in 0.1 s, runs 0.3 s, decelerates in 0.1 s, and dwells 0.5 s, against a constant load of 0.1 N.m. The tool reflects the mass and the screw to 1.01 kg.cm2, which against the 0.20 kg.cm2 rotor is a 5.0:1 ratio, marked good. The angular acceleration is 3,142 rad/s2, the acceleration torque is 0.38 N.m, the peak is 0.48 N.m, and the RMS is 0.18 N.m. The lesson: the moved mass and the screw body split the load inertia between them, and at 5:1 the axis tunes cleanly with room to spare on the torque.

Example 2: a belt and pulley with a reducer

Here a belt carries a heavy load, and a reducer keeps it tunable. A 40 kg mass moves on a 60 mm pulley, through a 10:1 gear ratio, driven by a motor with a 0.8 kg.cm2 rotor. The move goes to 1500 rpm in 0.15 s, runs 0.4 s, decelerates in 0.15 s, and dwells 0.3 s, against a 0.4 N.m load. The mass reflects through the pulley radius and then through the square of the 10:1 ratio to 3.60 kg.cm2, which against the 0.8 kg.cm2 rotor is a 4.5:1 ratio, marked good. The acceleration torque is 0.46 N.m, the peak is 0.86 N.m, and the RMS is 0.42 N.m. The lesson: a 10:1 reducer pulls a heavy belt load, which would swamp the motor directly, down to a tunable 4.5:1, because the ratio cuts the reflected inertia by its square.

Example 3: a rotary index table

A known load inertia comes straight in. A rotary index table has a load inertia of 8 kg.cm2 direct on the motor at a ratio of 1, driven by a motor with a 1.0 kg.cm2 rotor. The move goes to 500 rpm in 0.15 s, runs 0.2 s, decelerates in 0.15 s, and dwells 0.1 s, against a 0.2 N.m load. The reflected inertia is the load inertia itself, 8.00 kg.cm2, and against the 1.0 kg.cm2 rotor that is an 8.0:1 ratio, within the 10:1 limit but not tight. The acceleration torque is 0.31 N.m, the peak is 0.51 N.m, and the RMS is 0.29 N.m. The lesson: 8:1 is acceptable for a simple index that only has to reach position, but it is too loose for a high-response contouring axis, which would want a reducer to pull it toward 3:1 to 5:1.

Example 4: too much inertia, fixed by a reducer

This shows the reducer as the fix for a runaway ratio. A 20 kg.cm2 load sits direct on a 0.5 kg.cm2 rotor at a ratio of 1, which gives a reflected inertia of 20.00 kg.cm2 and a ratio of 40.0:1, far too high. At that ratio the acceleration torque is 6.44 N.m and the RMS is 3.22 N.m, and the axis will oscillate no matter how it is tuned. Add a 3:1 reducer and the reflected inertia falls to 2.22 kg.cm2, about a ninth, so the ratio drops to 4.4:1, marked good, the acceleration torque falls to 0.86 N.m, and the RMS falls to 0.43 N.m. The lesson: reflected inertia falls with the square of the ratio, so a 3:1 reducer cuts it about nine times, which is why a reducer, not a bigger motor, is the standard fix for a high inertia ratio.

Example 5: a fast pick-and-place

A light load and a small motor give a tight, fast axis. A pick-and-place moves 5 kg on a 5 mm lead screw of 0.3 kg and 12 mm diameter, at a ratio of 1, driven by a tiny motor with a 0.05 kg.cm2 rotor. The move goes to 4000 rpm in just 0.05 s, runs 0.1 s, decelerates in 0.05 s, and dwells 0.2 s, against a 0.05 N.m load. The reflected inertia is only 0.09 kg.cm2, which against the 0.05 kg.cm2 rotor is a 1.7:1 ratio, tight enough for a high-response axis. The angular acceleration is a fierce 8,378 rad/s2, but because the inertia is so small the acceleration torque is only 0.11 N.m, the peak is 0.16 N.m, and the RMS is 0.07 N.m. The lesson: a light load and a small motor give a tight ratio and a fast, crisp axis, and the very short accel time drives a high angular acceleration without demanding much torque, because there is so little inertia to accelerate.

Three expert tips

Check the inertia ratio first

A servo controls the motor, not the load, so the inertia ratio is the test to run before you look at torque. Keep the reflected load inertia within about 10:1 of the rotor for general motion, and nearer 3:1 to 5:1 for a crisp, high-response axis. Above that the load dominates the motor, the axis overshoots, and it is hard to tune no matter how much gain you throw at it. The fix is a reducer, not a bigger motor, because a reducer cuts the reflected inertia by the square of the ratio, so a 3:1 reducer cuts it about ninefold. Size that reducer with the gear ratio tool, then come back and confirm the torque. Fixing the ratio first often makes the torque question easy, because a smaller reflected inertia means a smaller acceleration torque too.

Size for both RMS and peak

The two torque numbers test two different failures, so you have to clear both. The rated, or continuous, torque must clear the RMS across the whole cycle so the motor does not overheat over a long run, and the peak torque must clear the acceleration torque so it does not stall on the ramp. A motor can pass one test and fail the other: a short hard move can spike high on the peak while staying modest on the RMS, and a long steady move can be gentle on the peak while running the RMS up. Check both against the maker’s torque-speed curve, not the flat catalogue figure, because both ratings droop as the speed rises and a motor that clears the peak at low speed can fall short at the top of a fast move.

The move profile is half the answer

The load sets one half of the torque, and the move sets the other. A shorter acceleration time raises the angular acceleration in direct proportion, so it raises the peak and the RMS together, while a longer dwell lowers the RMS without touching the peak because it adds rest time to the cycle. This means you can trade the profile against the motor. Measure the real duty cycle before you size, because a short hard move with long dwells can carry a modest RMS despite a high peak, and sizing on the peak alone would leave you paying for a motor far bigger than the heating actually needs. Match the move to the job, then size to the RMS and peak that move really produces.

Limits of the method

This calculator gives a sound first pass at a servo axis, not a finished motion-control design. It reflects the load inertia through the mechanism and the ratio, computes the inertia ratio, builds a trapezoidal move, and works the torque through accel, run, decel, and dwell to get the peak and the RMS, which is most of what a sizing exercise needs to pick a servo and a gearhead from a catalogue. It treats the mechanism as rigid, the load torque as constant across the move, the trapezoidal profile as the real motion, and the efficiency of the mechanism as ideal. Those assumptions are close to reality for a well-built axis running in its normal range.

What it does not do is the detail a full design carries. It does not read the maker’s torque-speed curve, so it will not tell you whether the rated and peak torque hold up at the speed your move runs at, which you must check by hand against the curve. It assumes a rigid coupling, so it does not model the resonance a compliant belt or a long screw can introduce, which can limit the tuning even when the inertia ratio looks fine. It treats the mechanism as lossless, so it does not add the friction and preload torque a real screw or a real seal carries, which you should fold into the constant load torque. It does not size the drive, the cabling, the regeneration during braking, or the thermal duty of the drive electronics. Use the result to fix the inertia ratio and size the steady torque, then confirm the torque-speed curve, the coupling stiffness, the friction, and the drive against the manufacturer’s data and a qualified engineer before you commit to a motor.

Common mistakes to avoid

The first mistake is sizing on torque alone and ignoring the inertia ratio. A motor with plenty of torque can still be impossible to tune if the reflected inertia dwarfs the rotor, because the servo controls the motor and only feels the load through the coupling. Always check the ratio first, and fix it with a reducer before you worry about torque. The second is forgetting the screw body. On a ball screw the moved mass is not the only inertia; the screw itself is a spinning cylinder with its own inertia, and on a long or heavy screw it can be the larger term. The tool adds it for you, but if you reflect the mass by hand and skip the screw you will undersize the axis.

A third mistake is sizing on the peak torque alone and ignoring the RMS, which leaves a motor that never stalls but slowly overheats across a long run of cycles. Always check the continuous rating against the RMS as well. A fourth is reading the flat catalogue torque instead of the torque at the move speed, since both ratings droop as the speed climbs and a fast axis can fall short at the top of its move. A fifth is treating the reducer ratio as a linear lever on the inertia when it is a square one: a 3:1 reducer cuts the reflected inertia to a ninth, not a third, which is why a modest ratio does so much. Check the ratio first, add the screw body, size for both RMS and peak, read the torque at the move speed, and remember the ratio squares, and the numbers the tool gives will match the axis you build.

Frequently asked questions

Sources, disclaimer, and editorial transparency

The servo sizing relationships used here follow standard motion-control practice. The reflected inertia of a ball-screw load is the mass times the lead over two pi, squared, plus one half the screw mass times the screw radius squared, and a belt load is the mass times the pulley radius squared, each divided by the square of the gear ratio; a known load inertia through a reducer is the load inertia divided by the ratio squared. The inertia ratio is the reflected load inertia over the rotor inertia, held at or below 10:1 for general motion and 3:1 to 5:1 for high-response axes. The angular acceleration is 2 pi times the speed change in rpm over 60 times the accel time, the acceleration torque is the combined inertia times the angular acceleration, and the RMS torque is the square root of the time-weighted sum of the segment torques squared over the total cycle time. Unit constants are 1 kg.cm2 equals 1e-4 kg.m2. This calculator and guide are built and reviewed by the OpsCalculators team; see our Editorial Policy for how each tool is researched, built, and tested.

Results are accurate estimates for planning and education, not a substitute for a full motion-control design or an engineering review. The method treats the mechanism as rigid and lossless, the load torque as constant across the move, and the trapezoidal profile as the real motion, but it does not read the manufacturer’s torque-speed curve, so it will not confirm the rated and peak torque hold up at your move speed. It does not model the resonance a compliant belt or a long screw can introduce, does not add the real friction and preload torque of the mechanism, and does not size the drive electronics, the regeneration during braking, or the thermal duty of the drive. Confirm the torque-speed curve, the coupling stiffness, the mechanism friction, and the final motor and drive selection against the manufacturer’s data and a qualified engineer before you commit to a servo. See our full Disclaimer. OpsCalculators.com is operated by MAFHH INTERNATIONAL LTD. Your inputs are processed in your browser and are never stored; see our Privacy Policy.