Electromechanical Actuator Load Locus Calculator
Electromechanical Actuator Load Locus Calculator
Size a linear electromechanical actuator against a sinusoidal command. The page sweeps one whole cycle, adds up the five torques the actuator has to fight, reflects them through the screw to the motor shaft, and plots the trajectory of motor torque against motor speed — the load locus. The headline is whether that locus fits inside the motor and drive’s envelope and by how much at its worst point, with the five contributions at that point broken out, because reflected inertia is usually the one that decides the answer.
Actuator load locus
a maxon EC-i 52 180 W on a 48 V bus through a drive limited to 8 A, driving a 16 × 5 mm ball screw at 90% efficiency, the nut working a link of 150 mm radius, and the link swinging a 400 × 300 × 10 mm steel plate hinged along its short edge and held 20° off vertical, through ±10° at 2.2 Hz with a 20% margin
One sinusoid in, one sinusoid out
Tload = J·δ̈ + B·δ̇ + K·δ + Toff + Ta J = JL + n²·JR B = n²·Bm
Tmargin = (1 + M)·Tload
n = 2πL ÷ p F = Tmargin ÷ L Tr = F·p ÷ (2π·η) = Tmargin ÷ (n·η)
ωm = n·δ̇ v = L·δ̇ I = Tr ÷ Kt
T(ωm) = Kt·(V − Ke·ωm) ÷ R truncated at Kt·Ipeak
the locus fits when |Keωm + R·I| ≤ V everywhere, and that expression is itself a sinusoid, so
Vmin = √(α² + β²) + |γ| with α = Kenk δmaxω + gQ, β = gP, γ = gC, g = R(1+M) ÷ (Ktnη)
- n
- motor radians per radian of load travel, 2πL ÷ p. It assumes the link’s effective moment arm L is constant through the stroke, which a real bellcrank’s is not
- JR
- everything that turns at motor speed — rotor, coupling, screw shaft, nut, bearings. Reflected to the pivot it becomes n²·JR, which for a screw drive is usually the biggest single term in the whole calculation
- B
- reflected viscous damping, n²·Bm. Damping reflects by the square of the ratio for exactly the reason inertia does: the motor’s own damping torque is Bm·n·δ̇, and turning a motor torque back into a load torque multiplies it by n a second time
- η
- screw driving efficiency. It divides the torque and does not appear in the speed. Back-driving efficiency is a different and smaller number
- k
- trackability factor, between 0.707 and 1. 0.707 is the −3 dB amplitude the actuator actually achieves at its bandwidth; 1 is the worst case that keeps margin in the motor
- P, Q, C
- the load torque written as P·sin ωt + Q·cos ωt + C, with P = kδmax(K − Jω²), Q = B·kδmaxω and C = Kδ0 + Toff + Ta. Every figure on this page is a closed form in these three
Worked example
a maxon EC-i 52 180 W on a 48 V bus through a drive limited to 8 A, driving a 16 × 5 mm ball screw at 90% efficiency, the nut working a link of 150 mm radius, and the link swinging a 400 × 300 × 10 mm steel plate hinged along its short edge and held 20° off vertical, through ±10° at 2.2 Hz with a 20% margin
The ratio is n = 2π × 150 mm ÷ 5 mm = 188.50 motor radians per load radian. The rotor's 17 mg·m² and the screw's 15.1 mg·m² therefore appear at the pivot as n²·JR = 1.141 kg·m², beside the plate's own 502.4 g·m² — so the total is 1.643 kg·m² and the motor and screw are 2.27 times the load
The plate is 9.42 kg at 7,850 kg/m³, so m·b²/3 = 502.4 g·m² about the hinge; its weight of 92.38 N acting at r·sin 20° = 0.0684 m gives a constant 6.319 N·m, and m·g·r·cos 20° = 17.36 N·m per radian is the restoring stiffness gravity provides about that held position
Collapsing the cycle: P = kδmax(K − Jω²) = −51.760 N·m, Q = 0 because no viscous damping was entered, and C = 6.319 N·m. The peak load torque is therefore √(P²+Q²) + |C| = 58.08 N·m at the extreme of travel, where the speed is zero
At that worst point the five contributions are: reflected inertia 38.04 N·m, the plate's own inertia 16.75 N·m, the spring −3.03 N·m, the constant offset 6.319 N·m and nothing from viscous friction or lateral acceleration — so reflected inertia is 65.5% of the load torque
With the 20% margin that is 69.69 N·m at the pivot, 464.6 N in the screw, and Tmargin ÷ (n·η) = 410.8 mN·m at the motor — 4.47 A. The peak motor speed is n·kδmaxω = 4,343 rpm
The envelope at the worst point is the peak current limit, Kt·Ipeak = 735.2 mN·m, so the locus fits with 324.4 mN·m in hand. The load line is much further away — 1.971 N·m at its closest — and the RMS torque is 262.7 mN·m, 62.3% of the 421.8 mN·m continuous rating
The inverse answers: the locus still fits on a bus of 41.91 V — 87.3% of the 48 V entered — and at this amplitude the highest command frequency the motor can track is 2.52 Hz, so 2.2 Hz is 87.3% of it. Peak electrical power is 100.6 W (2.097 A from the bus) and 11.71 J per cycle comes back out again
The five torques, and which of them you can usually ignore
| Term | What it is | How it scales | What it does to the locus |
|---|---|---|---|
| Inertia, J·δ̈ | the load’s inertia about the pivot plus everything rotating at motor speed, reflected by n² | with the square of the command frequency, and with amplitude | peaks at the extremes of travel, where the speed is zero — so it sets the top-left corner of the locus, and it is almost always the term that sizes the motor |
| Viscous friction, B·δ̇ | the motor’s own damping constant, reflected by n² | linearly with frequency and amplitude | peaks at the null crossing, in quadrature with inertia, so it tilts the locus rather than raising its corner. Usually small, and usually not on the datasheet |
| Spring, K·δ | the restoring stiffness of a seal, a hinge, a spring return, or gravity on a load hanging below its pivot | with amplitude only — not with frequency | opposes inertia, because both go as sin ωt. At ω² = K ÷ J they cancel exactly and the actuator needs almost no torque; that is the rigid-body resonance and it is not the one that limits a real actuator |
| Constant offset, Toff | a thrust line offset, a process pressure, a belt tension, gravity on an off-axis mass | not at all | shifts the whole locus, so the two halves of the cycle stop being mirror images and one extreme demands much more than the other |
| Lateral acceleration, m·a·d | the load’s own mass reacting an acceleration of the whole assembly | with the acceleration you design for | another constant, and it adds to the one above — which is why a design that is comfortable on the bench can be marginal in service |
What this model does not contain
| Left out | Why it matters | What to do about it |
|---|---|---|
| Structural compliance | this is a rigid-body model. A real actuator’s load path — screw, housing, bearings, link, backup structure — is a spring, and the moving mass on the far end of it resonates. That resonance is usually well below the frequency this page says the motor can track, and it, not the motor, sets the achievable bandwidth | measure or model the load-path stiffness and the resonance it makes with the moving inertia, and treat the frequency on this page as an upper bound you will not reach |
| Backlash and lost motion | a sinusoidal command reverses twice a cycle, and every reversal crosses whatever lost motion the screw, nut and link joints have. It costs position accuracy and it excites the structure | preload the screw and the joints, and judge accuracy from a measured reversal test, not from this page |
| The control loop | there is no loop here at all. The command is assumed to be followed exactly, which is what the trackability factor k is a crude stand-in for. A real loop has finite gain, phase lag and a current regulator with its own bandwidth | use the trackability factor honestly, and check loop stability separately |
| Thermal transients | the RMS check on this page compares a cycle’s RMS torque with a continuous rating. That is only fair if the cycle is short compared with the motor’s thermal time constant, and it says nothing about the winding’s short-term rise | check the cycle time against the winding and housing time constants on the datasheet |
| Load-dependent screw efficiency | efficiency varies with load, lead angle, preload and lubrication. A single catalogue number is an approximation, and the figure for back-driving is different again | run the page at the top and bottom of the efficiency range your screw supplier will commit to and see how much the answer moves |
| The efficiency applied to the rotor’s own inertia | Tr = Tmargin ÷ (n·η) divides the whole margined torque by η, including the part that came from the rotor’s own inertia — which never passes through the screw. The standard sizing convention does this and it is conservative by 1 ÷ η on that one term | nothing, unless you are chasing the last few per cent; then account for the rotor’s inertia torque separately |
Reading the locus, and why the rotor is usually the load
An electromechanical actuator that follows a sinusoidal command does not sit at one point on the motor’s torque-speed plane; it traces a closed curve round it, twice per cycle. That curve is the load locus, and sizing the actuator means asking one question about it: does the whole curve fit inside what the motor and its drive can do, and by how much at its worst point? A single-point calculation cannot answer that, because the demanding instants are in different places. Torque peaks at the extremes of travel, where the speed is zero, because that is where the acceleration is greatest. Speed peaks at the null crossing, where the acceleration is zero and the only torque left is whatever constant load the actuator works against. Put a static offset into the command and the spring term no longer vanishes at the null either, the two halves of the cycle stop being mirror images, and the locus can double back on itself — which is why the chart places every point at its own speed rather than spacing them evenly.
The reflected inertia is usually the answer. A screw drive is a very high ratio: 2πL ÷ p, which for a 150 mm link and a 5 mm lead is 188 motor radians for every radian of load travel. Rotating inertia reflects to the pivot by the square of that, so 321 g·cm² of rotor and screw — a number you would ignore on its own — becomes 1.14 kg·m² at the pivot, more than twice the load it is driving. On this page’s defaults, reflected inertia is 65.5% of the load torque at the worst point, and the plate the actuator exists to move is a minority shareholder. That is not a quirk of the example: above the inertia-matched ratio, n² = JL ÷ JR, the machine is mostly accelerating itself, and a screw drive sized for force is almost always above it. If the motor is too small, a bigger one with a bigger rotor can make the problem worse.
Two limits bound the locus and they behave completely differently. The peak torque limit is a flat ceiling at Kt times whatever current the drive and the winding will allow. The load line, T = Kt(V − Keω) ÷ R, is a straight line from a stall torque of KtV ÷ R down to zero at the no-load speed V ÷ Ke — and for a low-resistance servo motor it is steep. This example’s stall torque is more than twenty times the peak the drive will permit, so in the whole speed range the locus visits, the envelope is simply the current limit and the load line never comes into view. That does not mean the voltage is irrelevant: the minimum bus voltage at which the locus still fits is reported above, and it is the number that sizes the battery. What it means is that the voltage bites in frequency, not in torque. Torque demand grows as the square of command frequency while speed demand grows linearly, but the load line is so steep that the last few per cent of no-load speed are consumed abruptly — which is why the highest trackable frequency on this page is often set by the voltage even when the torque limit looks like the closer of the two.
Peak torque and RMS torque answer different questions. The peak has to fit under the peak limit or the drive current-limits and the actuator stops following the command. The RMS has to fit under the continuous rating, because winding heating follows I²R and RMS current is what it integrates — and for a sinusoid the RMS of the varying part is 1 ÷ √2 of its peak, so a locus whose peak is comfortably inside the envelope can still cook the motor if the constant part of the load is large. Both numbers are above, with the RMS as a percentage of the continuous rating. For the inverter’s own losses at these currents use the BLDC inverter loss calculator, and for the envelope of the machine on its own, with a generic load on the same axes, the BLDC torque-speed envelope calculator, which now draws the same load line at a stated bus voltage so the two pages can be read against each other.
Half the cycle is usually regeneration. Whenever the load torque and the direction of travel disagree, the machine is braking: torque one way, speed the other, power flowing out of the motor and into the DC link. The locus on the chart is drawn signed, so the loop passes through all four quadrants, and the arcs in the second and fourth — positive torque at negative speed, or negative torque at positive speed — are the braking ones. On this page’s defaults that is 49% of the cycle and 11.71 J every cycle, which at 2.2 Hz is 25.75 W of average returned power with a peak of 94.83 W. A bus that cannot absorb that will simply rise until something trips, so it has to go somewhere: into a battery that will take it, into link capacitance big enough to swallow it, or into a brake resistor. The regenerative braking calculator sizes that. The figures here assume a lossless inverter, so they are the optimistic end of what actually arrives at the link. The peak electrical power and the returned energy are the only two figures on this page that are not closed forms — the peak needs the root of a quartic and the energy needs that root’s position — so both are evaluated at the 72 points round the cycle that the table under the chart lists, and the peak reported is exactly the largest number in that table.
What this page is not. It is a rigid-body sizing model. There is no structural compliance in it, no backlash, no control loop and no thermal transient. A real actuator’s load path is a spring, and the moving mass on the end of it resonates — usually at a frequency well below anything this page predicts, and that resonance, not the motor, is what sets the bandwidth you will actually achieve. Gimballed actuator installations are studied precisely because that compliance couples the actuator to the structure it is mounted on. Treat the highest trackable frequency here as an upper bound that a real installation will not reach, and the margin as arithmetic, not as reassurance.
Frequently asked questions
Why does the chart put each point at its own speed instead of spacing them evenly?
Because the speed goes as cos ωt, so equal steps in time are very unequal steps in speed: the samples bunch up at the extremes of travel and thin out through the middle. And with a static offset or a constant load torque the trajectory doubles back on itself, visiting the same speed twice at different torques. Spacing the points by their position in the list rather than by their speed would draw a completely different curve — a circle comes out as a sine wave. The plot box is not square, so the locus is stretched horizontally; that does not matter here because the two axes are different quantities.
Should I use a trackability factor of 0.707 or 1?
1, unless you can justify otherwise. 0.707 is the −3 dB point: at the frequency where a closed-loop actuator’s response has fallen 3 dB it is only following 70.7% of the commanded amplitude, and sizing at 0.707 sizes for what the actuator will really do at its bandwidth. 1 sizes for the amplitude that was commanded, which is the worst case, and the difference is a factor of 1.41 in speed and in torque — so it is not a detail. Sizing at 1 keeps margin in the motor; sizing at 0.707 spends it.
Is the reflected inertia really bigger than the load?
On this page’s defaults, more than twice as big. Reflected inertia is n²·J_R, and for a screw drive n is in the hundreds, so n² is in the tens of thousands. The crossover is the inertia-matched ratio, where n² equals the load inertia divided by the rotating inertia; above that the motor spends most of its torque accelerating itself, and a screw drive picked for force rather than for inertia matching is almost always above it. It is the reason fitting a bigger motor sometimes makes an actuator slower rather than faster, and the reason the screw shaft’s own inertia — as much as the rotor’s, here — is worth looking up rather than guessing.
Why is the load line nowhere near my locus?
Because a low-resistance servo motor has an enormous stall torque. K_t·V ÷ R with this motor on 48 V is over 15 N·m, more than twenty times the peak torque the drive will let it make, so between standstill and the no-load speed the load line towers over everything. It only becomes the binding limit when the peak speed gets within a few per cent of the no-load speed, and then it bites very suddenly. The number that tells you where you stand is the minimum bus voltage above, and the one that tells you how much frequency you have left is the highest trackable frequency.
What is the difference between driving and back-driving efficiency?
The friction torque in the screw is roughly the same whichever way the power flows, but it subtracts from the output in both directions. Driving, the motor must supply the ideal torque plus the friction, so T = F·p ÷ (2πη). Back-driving, the load delivers the ideal torque minus the friction, so the efficiency comes out as 2 − 1 ÷ η, which for a 90% screw is 88.9% and sits in the 0.8–0.9 range ball screw suppliers quote. Below 50% driving efficiency that expression goes negative, which is exactly what self-locking means: the load cannot drive the screw at all and the motor has to supply torque even to let the load down.
Will my screw back-drive, and does it matter?
A ball screw almost always will: at a 90% driving efficiency the back-driving efficiency is positive, so a load left on the actuator with the motor unpowered will run the screw backwards. That matters for anything that has to hold a position without power, and it is why such actuators carry a brake. The old rule of thumb that a lead under a third of the screw diameter will not back-drive comes from sliding screws and does not transfer: the example here has a lead exactly 0.3125 of its diameter and back-drives freely, because a ball screw’s friction coefficient is an order of magnitude lower.
Can I trust the highest trackable frequency as a bandwidth figure?
No, and it is the most misused number on the page. It is the frequency at which the rigid-body locus first touches the motor’s envelope at the amplitude you asked for — a torque-and-voltage limit, nothing more. A real actuator’s bandwidth is set by the resonance between the compliance of its load path and the mass on the end of it, by the current regulator’s own bandwidth, and by how much phase margin the position loop can afford. That resonance is usually well below this figure. Use it as an upper bound that tells you when the motor is definitely not the problem.
Where does the design margin come from?
From whoever is signing the design off. It is a convention, not a standard: there is no published figure that makes 20% correct. It exists because the load inertia, the friction, the seal stiffness and the efficiency are all known less well than the arithmetic suggests, and because the winding will be hotter than the datasheet’s 25 °C. Applying it to the load torque before reflection, as this page does, is the usual convention. Put your own programme’s number in and take the answer as arithmetic rather than as a safety factor.
Related calculators
References
- maxon. EC-i 52 Ø52 mm, brushless, 180 W, with Hall sensors — product data for part 579165: 48 V nominal, 4,950 rpm no load, 423 mN·m nominal torque at 4.59 A, 0.284 Ω terminal resistance, 91.9 mN·m/A torque constant, 104 rpm/V speed constant, 170 g·cm² rotor inertia, 6,000 rpm maximum permissible speed. Fetched and read; every motor default on this page is from this sheet.
- maxon. Motor data and operating ranges (maxon academy / catalogue technical pages). Defines the torque constant kM and speed constant kn and states their relation: kM·kn = 1 in SI units, and 30,000 ÷ π = 9,549 in the catalogue’s mN·m/A and rpm/V — which is the identity Kt = Ke this page checks the reader’s pair against. It also defines the continuous and short-term operating ranges.
- THK. Features of the Ball Screw — Driving Torque One Third that of a Sliding Screw, technical document en_b15_006. Gives the driving torque as T = Fa·Ph ÷ (2πη₁) and the reverse (back-driving) torque as T = Ph·η₂·Fa ÷ 2π, with a worked example at η₁ = 0.96 for a ball screw against 0.32 for a sliding screw. This is the reflection the page uses.
- maxon. EPOS4 Compact 50/8 positioning controller data: 10–50 V DC, 8 A continuous output current, 30 A peak for 5 s. The source of the 8 A peak current limit in the worked example, and of the point that a continuously repeating cycle must be sized on the continuous rating.
- Linear Motion Tips. How to determine if a screw will back drive. Quotes reverse efficiency η₂ of 0.8 to 0.9 for ball screws, and the rule of thumb that a lead under a third of the screw diameter resists back-driving — a rule that comes from sliding screws and that the ball screw in this page’s example does not obey.
- Roton Products. Screw Backdriving Efficiency. States that a negative back-driving efficiency is what self-locking means — at an 8° lead angle the back-drive efficiency is −13%, so drive torque is needed even to lower a load — which is the behaviour the 2 − 1 ÷ η expression reproduces below 50%.
- Orr J S, Wall J H, Barrows T M. Simulation-Based Analysis and Prediction of Thrust Vector Servoelastic Coupling. NASA Technical Reports Server, accession 20230009355. On compliance in a gimballed actuator’s load path, the resonance it forms with the moving mass, and how that resonance rather than the actuator’s own capability governs the achievable response — the reason the rigid-body frequency figure on this page is an upper bound.