Inertia Ratio and Servo Oscillation: Causes, Limits and Fixes

Key Takeaway

Inertia ratio servo oscillation is a number problem before it is a tuning problem. Keep the ratio under 10:1 for general work and under 5:1 for fast acceleration, and when the axis still rings, look at shaft stiffness and structural modes, not at the gain.

1. What Inertia Ratio Actually Is

Servo oscillation on a machine axis is one of the most frustrating faults to diagnose, because it sounds like a tuning problem and it is usually a numbers problem. The drive is asked to accelerate a mass it does not know about. It applies the torque it thinks is right, the mass overshoots, the drive applies more torque, and the loop starts hunting. Raising the gain makes it worse. Lowering the gain makes the axis soft. Neither one addresses the actual cause. Inertia ratio servo oscillation is a numbers problem before it is a tuning problem, and the numbers are on the datasheets of the motor and the load.

Inertia ratio is the number that describes the problem. It is the ratio of the load inertia reflected to the motor shaft to the motor’s own rotor inertia:

inertia ratio = J_load / J_motor

Every servo drive has a recommended limit, and it is not negotiable in the way that a speed limit is. The usual limits are:

Application	Recommended inertia ratio	Why
General purpose, low acceleration	Under 10:1	Comfortable tuning margin
High acceleration or high rigidity	Under 5:1	Stiffness without ringing
Very high response, direct drive	Under 3:1	Near-zero compliance in the loop
Absolute limit	30:1	Above this, no gain setting is stable

A motor with a rotor inertia of 0.31e-4 kg m^2 driving a load of 2.5e-3 kg m^2 reflected to the motor gives a ratio of 8.1:1. That motor is comfortable. The same motor driving a load of 7.5e-3 kg m^2 gives 24.2:1, and the axis will hunt, buzz, and get worse as the acceleration rises.

2. Calculating the Inertia Ratio

The calculation has three parts: the motor rotor inertia from the motor datasheet, the load inertia at the load, and the transformation that reflects the load inertia to the motor shaft. Inertia ratio servo oscillation has one definition and one test, and once you have both, the cause is usually obvious.

Motor rotor inertia

Inertia ratio servo oscillation is a numbers problem before it is a tuning problem, and the numbers are on the datasheets of the motor and the load. This is a datasheet number. A typical 750 W AC servo motor has a rotor inertia of about 0.31e-4 kg m^2, a 1 kW motor about 0.52e-4 kg m^2, and a 2 kW motor about 1.9e-4 kg m^2. Note that the rotor inertia rises roughly with the square of the motor size while the rated torque rises linearly, so a bigger motor improves the inertia ratio in both directions at once.

Load inertia, in the form it is easiest to compute

Load element	Inertia at its own axis
Solid disc or plate, mass m, radius r	J = m x r^2 / 2
Thin ring or pulley, mass m, radius r	J = m x r^2
Solid cylinder, mass m, radius r	J = m x r^2 / 2
Long rod about one end, mass m, length L	J = m x L^2 / 3
Long rod about its centre, mass m, length L	J = m x L^2 / 12
Rectangular plate about an edge	J = m x (L^2 + h^2) / 3

For a table of mass m and dimensions a by b, rotating about the centre: J = m x (a^2 + b^2) / 12. The rotor inertia is the denominator of the whole calculation, and inertia ratio servo oscillation begins with a wrong rotor inertia more often than with a wrong load.

Reflecting the load to the motor

For a screw drive. A translating mass m moved by a screw of lead L (metres per revolution) behaves at the motor as a rotational inertia

The load inertia table is the set of formulae that turn a shape into a number, and inertia ratio servo oscillation cannot be diagnosed without them.

J_reflected = m x (L / (2 x pi))^2

Check the units: L / (2 pi) is the radius of an imaginary pulley with the same lead, so the formula is the m x r^2 of a point mass at that radius.

For a belt drive. With a motor pulley radius r1 and a driven pulley radius r2, the speed ratio is i = r2 / r1, and the reflected inertia is:

J_reflected = J_load / i^2

For a gear train. Same rule with the ratio squared:

J_reflected = J_load / i^2

A complete worked example

Step 1, the table as a translating mass: J_reflected = 25 x (0.020 / 6.2832)^2 = 25 x (0.0031831)^2 = 25 x 1.0132e-5 = 2.533e-4 kg m^2. Reflecting the load to the motor is the step that converts a mechanical description into an electrical one, and inertia ratio servo oscillation depends entirely on it.

Step 2, the screw assembly as a rotating mass: J = 3 x 0.03^2 = 2.7e-3 kg m^2, at the motor already, so 2.7e-3.

Step 3, the coupling: J = 0.5 x 0.025^2 = 3.13e-4 kg m^2.

Step 4, the total at the motor: 2.533e-4 + 2.7e-3 + 3.13e-4 = 3.266e-3 kg m^2.

Step 5, the motor: a 750 W servo with 0.31e-4 kg m^2.

Step 6, the ratio: 3.266e-3 / 0.31e-4 = 10.5:1.

That is just over the 10:1 limit for general work, and the axis is a typical general-purpose axis. The dominant term is the screw assembly at 2.7e-3, which is 83 percent of the total. A heavier screw is a bad trade: it costs inertia and it costs money. This is where the calculation starts to drive design decisions rather than just confirm them.

3. Reflected Inertia Through a Gearbox

A gearbox is the standard answer to a high inertia ratio, and the arithmetic is why it works so well. Calculating the inertia ratio takes ten minutes with a calculator, and it is the highest value ten minutes in diagnosing inertia ratio servo oscillation.

J_reflected = J_load / i^2

Take the 3.266e-3 kg m^2 axis above and put a 10:1 gearbox on it. The reflected inertia becomes 3.266e-3 / 100 = 3.266e-5 kg m^2, and the inertia ratio against the same motor becomes 3.266e-5 / 0.31e-4 = 1.05:1. The problem is gone.

But the gearbox is not free, and the reason is worth being precise about. The gearbox has its own reflected inertia at the motor, typically 3 to 10 times the motor rotor inertia for a strain wave gearbox. For our motor that is 0.93e-4 to 3.1e-3 kg m^2. Add it back:

Component	Inertia at motor (kg m^2)
Load through 10:1 gearbox	3.27e-5
Gearbox reflected inertia (typical, mid-range)	1.0e-3
Motor rotor inertia	0.31e-4
Total	1.063e-3
Inertia ratio	3.43e-4 / 0.31e-4 = 1.1:1 (load only) but 3.43:1 including the gearbox

So the gearbox took the load-driven ratio from 10.5:1 down to about 3.4:1 including the gearbox’s own inertia. The improvement is real but it is much smaller than the ratio squared suggests, and it is much smaller than the rule of thumb that says “put a gearbox on and the problem goes away”.

The gearbox also adds a torsional compliance between the motor and the load, and that compliance creates a resonance mode. The mode frequency is roughly the closed-loop bandwidth divided by the square root of the inertia ratio reduction, and with a big inertia reduction the mode can land at 30 to 60 Hz, which is right in the band where the position loop wants to run. This is why a badly chosen gearbox can make an axis worse rather than better, and it is covered in the harmonic drive gearbox selection guide.

4. The Limits and Why They Are Tight

The recommended limits are empirical and they come from the relationship between the inertia ratio and the phase margin of the velocity loop. The velocity loop is a proportional-plus-integral controller acting on a mechanical plant that is a pure inertia, and the closed-loop response of an inertia driven by a PI controller has a pair of lightly damped poles whose damping ratio falls as the inertia ratio rises. The recommended limits on inertia ratio servo oscillation are empirical, and tight enough to treat as acceptance criteria rather than guidelines.

In practical terms, at a ratio of 10:1 the axis tunes cleanly with the manufacturer’s recommended gain and has margin to spare. At 20:1 the recommended gain produces a lightly damped response and the axis needs its velocity gain reduced by 30 to 40 percent, which makes it soft and increases the following error. At 30:1 the loop is unstable at any gain that gives useful stiffness. Above 30:1 there is no setting that works, and the answer is mechanical.

What the ratio does to the tuning

• Velocity gain. The recommended velocity proportional gain scales roughly with the inverse of the inertia ratio. Halve the ratio and you can roughly double the gain before the same stability limit is reached.
• Following error. A higher ratio means more of the commanded acceleration has to be supplied by the loop rather than by feedforward, so the following error during a fast move grows.
• Disturbance rejection. Pushing against the axis by hand gets a softer response as the ratio rises, because the motor has more inertia to accelerate.
• Settling time. The damped mode that appears at high ratios is what makes the axis ring after a step, and it lengthens the settling time.

A useful check that needs no datasheet: if the axis rings after a rapid positioning move and the ringing frequency does not change when you change the speed, suspect the inertia ratio. If the frequency does change with speed, suspect a structural resonance instead.

5. Structural Resonance: The Other Oscillation

A structural resonance is a natural frequency of the machine itself: a long axis on a flexible base, a cantilevered table, a long screw with an overhang, a cross slide with a heavy carriage at the end. The drive’s command excites it, and the drive cannot see it because it is measuring the motor, not the load. Not every oscillation is an inertia ratio problem, and confusing the two sends the technician in the wrong direction for a week.

How to tell them apart

Not every oscillation is an inertia ratio problem, and confusing the two sends the technician in the wrong direction for a week. A resonance excited at one particular speed is the clearest signature there is. It is also the most easily confirmed: slow the axis down past the speed and the fault disappears. The complete example is the shortest path to the number, and inertia ratio servo oscillation is either confirmed or dismissed by it in ten minutes.

Fixes, in order of preference

1. Change the structure. Shorten the overhang, stiffen the casting with a rib, move the motor to the far end, replace a long unsupported screw with a supported one. This is the only fix that costs nothing in axis performance.
2. Change the speed. If the process allows, run faster or slower than the resonance. Cheap and often not available.
3. Add a notch filter in the drive. Most drives have a configurable notch filter, and setting it to the measured resonance frequency plus 10 percent attenuates the mode. This is the standard fix and it works. It costs phase margin elsewhere and it does nothing about the fatigue the resonance is causing in the structure.
4. Add a low-pass filter on the velocity loop. Blunter than a notch and it attenuates everything above the corner frequency. Acceptable when the resonance amplitude is small.

The recommended limits on inertia ratio servo oscillation are empirical, and tight enough to treat as acceptance criteria. The measurement that identifies the frequency is simple: run the axis slowly from standstill through the suspect speed range with a modest acceleration and record the following error. The resonance shows as a spike in the trace at a specific speed.

6. Torsional Compliance in the Drive Train

Compliance in the drive train produces a mode that looks like a tuning problem and is not one, and it sits right next to the inertia ratio in the causes list.

Every shaft, coupling and screw has a torsional stiffness k (in Nm per radian). When the motor accelerates the load, the motor side twists and the load side lags. The two inertias, connected by a spring, form a two-mass system with a resonant frequency:

f_res = (1 / 2 pi) x sqrt( k x (1 / J1 + 1 / J2) )

Work a real case. A rigid coupling with a stiffness of 5000 Nm per radian connects a motor with J1 = 0.31e-4 kg m^2 to a load of J2 = 3.266e-3 kg m^2:

f_res = (1 / 6.2832) x sqrt( 5000 x (1 / 0.31e-4 + 1 / 3.266e-3) )

= 0.1592 x sqrt( 5000 x (32258 + 306) ) = 0.1592 x sqrt( 5000 x 32564 ) = 0.1592 x sqrt(1.628e8) = 0.1592 x 12760 = 2032 Hz

2032 Hz is far above any control bandwidth and it is not a problem. The coupling is stiff enough for this axis. Now halve the coupling stiffness to 2500 Nm per radian because the coupling has a rubber insert:

f_res = 0.1592 x sqrt( 2500 x 32564 ) = 0.1592 x sqrt(8.141e7) = 0.1592 x 9023 = 1436 Hz

Still high. The two-mass resonance only becomes a problem when the stiffness is low enough to bring it into the 30 to 80 Hz band where the position loop lives. That requires k of roughly:

k = (2 pi f)^2 / (1/J1 + 1/J2) = (2 pi x 50)^2 / 32564 = 98,696 / 32564 = 3.03 Nm per radian

Three Nm per radian is a very soft coupling. A rubber spider coupling on a machine with a large load inertia can be that soft, and that is precisely when the hunting appears. The rule is that a flexible coupling on a high inertia axis inserts a mode that the position loop can excite, and the fix is a stiffer coupling or a lower position loop gain. The shaft coupling guide has the stiffness figures for the common types.

7. Inertia Ratio Servo Oscillation: What to Do When the Ratio Is Already Too High

Existing machines cannot be re-geared overnight, so there is a defined order of fixes for a high inertia ratio on a machine that is already built.

1. Check the number first

Measure the actual rotor inertia from the drive’s motor parameter, and compute the actual load inertia from the real machine, including the screw, the coupling, the fixture and the workpiece. Do not trust the number from the original design drawing; the machine has been modified. A surprising share of “high inertia ratio” faults turn out to be a fixture added two years after commissioning.

2. Add inertia feedforward

If the drive supports inertia feedforward, set it to the measured load inertia. This does not fix the stability problem, because the ratio is still high, but it removes most of the following error and it makes the axis feel much better while you work on the real fix.

3. Reduce the velocity gain

Reduce the velocity proportional gain until the axis is stable, then raise the position loop gain only as far as the new velocity gain allows. The tuning order and the bandwidth rule are in the servo tuning guide.

4. Fit a notch filter

If the oscillation is at a fixed frequency, set the drive’s notch filter to it. This is the standard field fix and it is the right one when the resonance amplitude is small enough that the fatigue is not a concern.

5. Lower the acceleration

This is the cheapest fix in the list and the one most often skipped. The oscillation amplitude scales with acceleration, so halving the acceleration limit halves the disturbance. On a cutting axis it costs cycle time. On a transfer axis it costs nothing.

6. Add a gearbox

The mechanical fix, and it needs a mounting plate, a coupling change and a re-tune. It is worth doing when the machine is already good in every other respect.

7. Fit a larger motor

Undersized motors are a real cause. Stepping up one size increases the rotor inertia as well as the torque, and the ratio improves. It is a surprisingly effective and surprisingly cheap fix on a marginal axis.

8. Reduce the load inertia

Replace the screw with a smaller one, shorten the overhang, move the motor closer to the load, or reduce the carriage mass. On a machine with a 2.7e-3 kg m^2 screw assembly, a lighter screw is the single biggest available improvement.

That is the order. Do the free things first, because they fix most cases, and they cost nothing but the time to try them.

9. Verifying the Fix

The standard move

An inertia ratio problem that is fixed should be provable, and two measurements do it. Define a move: a fixed distance at a fixed velocity with a fixed acceleration, repeated 10 times. Record the peak following error, the settling time and the position spread across the 10 runs. Before the fix, the settling time is long and the following error trace is ragged. After the fix, the settling time drops and the trace becomes a clean shape.

The inertia test

Run the axis at a constant speed with a step change in velocity command and watch the following error. On an axis with a good inertia feedforward, the following error during the acceleration is close to the theoretical value from the velocity loop lag:

e = a x t_lag

With 2 m/s^2 of acceleration and a 0.5 ms velocity loop lag, e = 2 x 0.0005 = 0.001 m = 1 mm. If the measured following error is much larger than that, the inertia feedforward is wrong or missing, and the drive is fighting an unknown mass. That is the same diagnosis as a high inertia ratio even when the ratio itself is acceptable.

Record it

Write the ratio, the peak following error, the settling time and the repeatability on the machine’s acceptance sheet, with the date and the name of the person who ran the test. The next engineer to touch this axis needs those four numbers more than they need the drawings.

The broader context on how these numbers fit into a complete axis is in the industrial motion control guide, and the mechanical side of the fix is in the harmonic drive gearbox selection guide.

Related Reading

• Industrial Motion Control: Complete Guide to Servo, Stepper and Axis Design
• Servo Tuning: Position, Velocity and Current Loops Explained
• Servo Drive Selection and Sizing: Power, Regeneration and Cable
• Harmonic Drive Gearbox Selection: Ratio, Repeatability and Life

Sources

1. Yaskawa, Sigma-7 Servo Drive Operation Manual (inertia ratio, auto tuning, notch filter)
2. Rockwell Automation, Servo Drive and Motor Installation and Tuning (inertia mismatch)
3. Nidec, Servo Motor Sizing Guide (rotor inertia values)
4. Harmonic Drive Systems, Technical Reference (reflected inertia and gear ratio)
5. ISO 9283, Machine tools: Test conditions for positioning accuracy and repeatability

Key Takeaways

• Inertia ratio servo oscillation is diagnosed by computing the ratio: the load inertia reflected to the motor divided by the motor rotor inertia. General purpose work wants under 10:1. Fast acceleration wants under 5:1.
• Reflected inertia through a gear ratio falls as the ratio squared. A 50:1 gearbox divides a 4 kg m^2 load inertia by 2500, taking it to 0.0016 kg m^2 at the motor.
• A gear ratio lowers inertia ratio but adds its own reflected inertia and a torsional resonance. The net gain is real but smaller than the ratio squared suggests.
• Oscillation that changes frequency with speed is mechanical resonance. Oscillation at a fixed frequency regardless of speed is electrical or gain related.
• A low pass filter on the velocity loop masks a resonance and adds phase lag, which costs stability margin. Use it only after the mechanical cause has been addressed.