A servo axis that was sized from the peak torque alone will move the load on the test bench and overheat in week three of production. The numbers that decide whether an axis lasts are the inertia ratio, the root-mean-square torque over the real cycle, and the ambient temperature inside the cabinet.
Every so often a machine comes to us with an axis that misbehaves in a way nobody can pin down. It positions correctly when it is tested slowly. It overshoots when it runs at production speed. After an hour of three-second cycles the drive trips on motor temperature, and the maintenance log records the trip as an intermittent fault of unknown cause. The motor is never undersized in the obvious way, because somebody did check that it could move the load. What was not checked is how hard it has to work, how often, and against how much inertia.
Sizing a servo axis is arithmetic, not opinion, and it takes about an hour with a datasheet. The inputs are the mass to be moved, the mechanics that connect it to the motor, the motion profile, the cycle time and the ambient temperature. What follows is the way we do it, with one example carried through, because the example makes the traps visible.
Start with the move, not the motor
Suppose a 40 kg carriage has to travel 300 mm in 0.5 s, then stay put for another 0.5 s while a gripper works, giving a one-second cycle. The simplest profile is a trapezoid split into equal thirds: accelerate for a third of the time, run at constant speed for a third, decelerate for a third. For that shape the peak velocity is one and a half times the average, so 1.5 x 0.3 m / 0.5 s = 0.9 m/s, and the acceleration is 0.9 divided by 0.167 s, about 5.4 m/s squared.
Those two numbers already decide the mechanics. With a ball screw of 10 mm lead, 0.9 m/s means 90 revolutions per second, or 5400 rpm, which is beyond the comfortable range of most screws of that size and close to the critical speed of a long one. With a 20 mm lead it is 2700 rpm, which is ordinary. Choosing the lead before the motor is therefore not a detail; it sets the speed the motor must reach and, through the same factor, the torque it must produce. The factor that converts between the linear and rotary worlds is the lead divided by two pi: for a 20 mm lead, 3.183 millimetres of travel per radian.
Inertia ratio: the number that decides how the axis feels
The load inertia seen at the motor shaft is the mass multiplied by the square of that factor: 40 kg x (3.183e-3 m/rad) squared, about 4.05e-4 kg m squared. The screw itself is not free: a steel screw 25 mm in diameter and 500 mm long weighs about 1.9 kg and contributes roughly 1.5e-4 kg m squared, a third as much again. Couplings, the pulley of a belt drive and the brake disc all add their share, and on belt axes the belt mass is part of the moving mass.
Compare the total, here about 5.6e-4 kg m squared, with the rotor inertia of the motor under consideration, say 1.2e-4. The ratio is 4.6 to 1. As a rule of thumb, a ratio up to about 5 to 1 gives a stiff, well-behaved axis with standard control settings; up to 10 to 1 is workable with a stiff coupling and careful tuning; beyond that the axis becomes hard to tune, because the control loop is trying to command a mass it can barely feel through a connection that is never perfectly rigid. Overshoot at speed and a tendency to hunt at rest are the symptoms, and no amount of gain adjustment fixes a mechanical ratio.
A gearbox changes the picture more than any other single choice, because it divides the reflected load inertia by the square of the ratio. A 5 to 1 planetary reduces our 5.6e-4 to 2.2e-5, turning a 4.6 to 1 ratio into something under 1 to 1, while multiplying the torque the motor sees and dividing its speed. The cost is backlash, which for a planetary is quoted in arcminutes and translates directly into lost motion at the tool, plus another component to align and lubricate. On a positioning axis with a tight tolerance we would rather use a direct screw drive and accept a higher ratio than introduce backlash into the loop.
Torque: three components, one sum
The torque required at any instant is the sum of three terms. The first is inertia times angular acceleration. Our angular acceleration is the linear acceleration divided by the same 3.183e-3 factor, about 1696 radians per second squared, and the total inertia including the rotor is 6.8e-4, so the acceleration torque is about 1.15 N m. The second is friction, which on linear guides is often modelled as a coefficient of about 0.01 times the normal load, here 3.9 N, contributing barely 0.01 N m through the screw, plus the drag of seals and the preload of the nut, which the screw maker states and which is frequently the larger of the two. The third is gravity, zero on a horizontal axis and dominant on a vertical one: the same 40 kg hanging on the screw needs about 1.25 N m to hold, all day, whether it is moving or not. The screw efficiency, typically 0.9 for a ball screw, divides into the driving terms and multiplies into any back-driving case.
A vertical axis therefore has two extra requirements that horizontal axes do not: a continuous holding torque that heats the motor even when nothing moves, and a brake, because a servo that loses power does not hold anything. The brake is a safety component when a person can be under the load, and it is sized for the static load with margin, not for dynamic braking.
The duty cycle, which is where sizing is actually won
Peak torque tells you whether the axis can move at all. What decides whether it survives is the root-mean-square torque over the whole cycle, including the time the axis stands still, because that is what heats the motor. In the example the axis accelerates at 1.15 N m for 0.167 s, coasts at almost nothing for 0.167 s, decelerates at about 1.10 N m for 0.167 s, and then rests for 0.5 s. The root-mean-square over the one-second cycle works out at about 0.65 N m.
The motor must therefore have a continuous rating comfortably above 0.65 N m, and a peak rating above 1.15 N m. We take 30 per cent margin on the continuous figure as a matter of habit, because a machine that runs at 98 per cent of its thermal limit on the day of handover has nowhere to go when a guide gets stiff, when a seal is replaced with a tighter one, or when the customer asks for a cycle a tenth of a second faster. That last request arrives on almost every project. If the dwell in our example shrinks from 0.5 s to 0.2 s, the cycle drops to 0.7 s and the root-mean-square torque rises to about 0.78 N m, a 20 per cent increase from a change that looks like it has nothing to do with the motor.
Ambient temperature belongs in the same calculation. Motor and drive ratings are stated at a given ambient, usually 40 degrees, and derate above it. Inside a closed cabinet in a Turkish or Spanish summer, 55 degrees is not unusual, and at that point a drive may be delivering 20 per cent less continuous current than its label suggests. This is the same heat load that decides the cooling of the panel, which is why we size the axis and the panel cooling with the same spreadsheet rather than in two separate departments.
Regeneration, resolution and the things that are not the limit
Every deceleration returns energy to the drive. Whether it matters depends on the mass, the speed and how often it happens: a small horizontal axis dumps its energy into the drive capacitors and nobody notices, while a heavy vertical axis lowering a load repeatedly will raise the DC bus until the drive trips on overvoltage unless a braking resistor is fitted and sized for the average regenerated power, not the peak. The calculation is the kinetic and potential energy per cycle divided by the cycle time, and it is usually a surprise to nobody except the person who left it out.
Encoder resolution, by contrast, is almost never the limit and gets far more attention than it deserves. A 17-bit encoder gives 131072 counts per revolution; with a 20 mm lead that is 0.15 micrometres per count. No machine of this type positions to anything close to that. What actually limits repeatability is the stiffness of the coupling, the backlash of any gearbox, the thermal growth of the screw over a shift, and the rigidity of the frame the whole thing is bolted to. When an axis will not hold a tolerance, the answer is in the mechanics nine times out of ten.
What we hand over
A sizing calculation that can be checked is part of the documentation on our machines: the move profile with its velocity and acceleration, the mechanics and the reflected inertia, the inertia ratio, the peak and root-mean-square torque against the chosen motor with the margins stated, the regenerated energy per cycle, and the cabinet ambient assumed. After commissioning we log the actual drive current over a full production cycle and compare it with the calculation, because that is the only way to find out whether the assumptions about friction were right. An axis sized this way is boring for years, which is the highest compliment a machine builder can pay to a motor.
— GANI Engineering engineering team