# Inertia ratio

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#### Guy H. Looney

I would like to get some feedback on the topic of inertia ratio. I have recently been trying to gather information as to the importance of this specification. For those of you unsure of what I
mean, let me elaborate:

The inertia ratio is equal to the reflected load inertia divided by the motor's rotational inertia. The specifics of the calculation are not
important for this conversation. There are many recommended guidelines to address this topic. The most common is the 10:1 or 5:1 rule-of-thumb. Some manufacturers state that a stepper
system should have less than a 5:1 ratio, while a servo can have a 10:1 ratio.

The question is why does inertia ratio matter? If the motor/drive combination can handle the torque, speed, & regen shouldn't that be enough? I have an opinion on this topic but am curious as to what others think.

Let me throw out another idea. Let's assume that it is critial to maintain a certain inertia ratio. Every motor manufacturer spec's out the rotational inertia of its motors. If you'll look carefully, you'll see that the inertia spec increases if an absolute encoder &/or brake is added to the motor. Why is this? Are we to assume that by increasing the mass of the motor shaft we can effectively increase the motor's rotational inertia? If this is the case, is there
a limit to how much we can increase it or a set of guide-lines stipulating how the mass must be attached? If your answer states that the brake &/or absolute encoder are contained w/ in the motor housing & that's the reason the increase is relavant, you need to look at the construction of the motor. The brake &/or absolute encoder are not subjected to the magnetic field...therefore
they could be added outside the housing & the effect should be the same (w/ respect to increased rotor inertia). That being said, why
can't we simply put a mass on the shaft of a motor (outside the casing) to increase its inertia.

Some have said that it's a question of compliance & rigidity. Assume the load could be pressed on the shaft & that the bearing structure of the motor is sufficient to handle radial & axial loadings. If the drive/motor can handle the torque, speed, & regen is the inertia ratio important? If so, at what point does it matter?

This subject has long been a topic of conversation within our industry. I would really appreciate any feedback that is deemed relevant. I hope that I receive qualtity feedback, and if you do not feel comfortable with this topic, sit back and enjoy the responses.

Sincerely,

Guy H. Looney
Sales Engineer

Regan Controls, Inc.
475 Metroplex Dr.
Suite 212
Nashville, TN 37211
phone: (615) 333-1940, ext. 322
fax: (615) 333-1941
[email protected]
www.regancontrols.com

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#### David Zishuk

Guy;

Let me first say that with any engineering appplication there is give and take - Regeneration is no exception.

>Let me throw out another idea. Let's assume that it is critial to maintain a certain inertia ratio. Every motor >manufacturer spec's out the
rotational inertia of its motors. If you'll look carefully, you'll see that the inertia >spec increases if an absolute encoder &/or brake is added to the motor. Why is this?

>Are we to assume that by increasing the mass of the motor shaft we can
effectively increase the motor's rotational >inertia? If this is the case,
is there a limit to how much we can increase it or a set of guide-lines
stipulating how >the mass must be attached? If your answer states that the
brake &/or absolute encoder are contained w/ in the motor >housing & that's
the reason the increase is relavant, you need to look at the construction
of the motor. The brake >&/or absolute encoder are not subjected to the
magnetic field...therefore they could be added outside the housing & >the
effect should be the same (w/ respect to increased rotor inertia). That
being said, why can't we simply put a >mass on the shaft of a motor
(outside the casing) to increase its inertia.

NO!

The ratio should be relative to the motor's rotor, not encoder or brake attached to the motor, that is part of the load. Because most manufactures (Yaskawa Included) do not give the rotor only. You can safely assume the
"standard motor", even though it includes an encoder in our case, is rotor inertia. If the Servo Motor's rotor is so light that that the encoder inertia is significant, then it should be shown separately. Even Yaskawa's tiny 10W sigma mini, still has a rotor inertia very much larger than the encoder it comes with.

>Some have said that it's a question of compliance & rigidity. Assume the
load could be pressed on the shaft & that >the bearing structure of the
can handle the >torque, speed, & regen is the inertia ratio important? If
so, at what point does it matter?

This is more complicated, and I can't give a full [perfect] answer.

1) Inertia and Bandwidth

Inertia is a low pass filter, the closer to 1:1 the better your bandwidth (Stopping and Starting time) will be. If you have a large system, and you want 1000Hz response from your load, 1:1 or less, you can forget about it. Smaller systems have quicker response. If you size a system, and you have the RMS. and Peak torque for the application, it will work.

2) Stored Energy

Any Rotating inertia has kinetic energy "stored" in the rotating load. This energy, is quantified by Ke = 1/2*J*w^2. Where J is the total system
inertia (in Kg-M^2), w is the rotational speed in rad/sec.

If you slow it down, then energy is removed from the system and has to go somewhere. In the case of the Servo Motor system, it is "regenerated" onto the DC BUSS- and starts to raise the BUSS voltage. The first device to "absorb" the energy are the BUSS capacitors. If the capacitors can absorb this extra energy- you are home free. The Got-ya is the CAP(s) may not absorb all of it, and the next step is to add more capacitors or "burn" it up with resistance. Some Servo vendors allow more capacitance to be added, Yaskawa prefers the resistor route.

3) Got the Power??

If the Servo Amp has built in resistor(s) to "burn" the regen energy that the caps cannot. You must then check if the resistors wattage is high enough. Both RMS and Peak. Wattage is En / time (sec). The decel time in seconds is used in this calculation. If the built in resistor is not
enough or is no resistor, then a provision for an external resistor should be given.

4) Resistors

One other little got-ya is the Ohmic value of the resistor: it should be low enough to quickly burn up the energy, but not so low to allow a large
destructive current through the regen switching circuit.

The Servo AMP manufacture should give guidelines for resistor Wattage, Ohmic value, and derating

5) Dynamic Brake Resistor

Another reason for the inertia ratio spec is because of DB resistors in Servo AMPS. These are the resistors that the motor leads are shorted to
when a power loss occurs. If you inertia is big, I mean really big, and you may just "POP" this normally small resistors. A good rule of thumb
(sorry this thumb thing is back) is your RMS. Regen Wattage exceeds the servo AMPs RMS capacity, there is a good chance the DB resistor will not handle the "E-STOP"

When a load is fighting gravity, and moves down (no up) Potential energy is changed to Kinetic and, in the case of Servo motor system, ends up on the DC BUSS as higher voltage. this must also be considered in sizing regen. Simple solution, use counterbalance.

7) Read this all more than once, tabulate your questions and call me with them.

David

#### PhilCorso

In response to Guy Looney's query, hopefully I can reply in general terms which do not cover a specific application, nor motor size:

As you already know inertia or moment of inertia or ratio, are not required for normal operating speed, i.e., the torque supplied by the motor equals the torque required by the driven machine, even if connected via a gear box, jack shaft, pulley and belts etc. In other words, if correctly sized, a motor's rated power output should correspond to the power absorbed by the driven machine.

Transient Operation
Important parameters are required to determine operational performance during speed changes, i.e., run-up, braking, reversing. They are the
mass (more importantly moment of inertia) of the motor, any additional significant masses (gear, drive shaft, coupling), the load torque during
run-up, the available torque produced by the motor during run-up, and the torque imparted to the entire drive-train during braking (where
required). Thus, the Equivalent Moment of Inertia of the drive-train is equal to GD^2 (Metric units) or equal to WR^2 (English units). It is
needed to determine the time to effect a desired speed change, and the amount of power to effect that change. The selected units are important as will be shown later in this discussion.

Speed Change
In general, every speed change is associated with a change in kinetic energy. This means that for a change in speed per unit time a certain amount of power is required.

Inertia Ratio
The moment of Inertia of each of the drive train elements having significant mass, must be referred or converted or, in Guy's terms,
"reflected", to the motor shaft speed. The rotating system's equivalent moment of inertia is calculated from the sum of the individual moments
of inertia of all the elements. This includes the motor, and each additional element converted to the motor speed using the "inertia ratio" as follows, in metric units:

GD^2(equivalent, referred to motor shaft speed, Nm) equals

GD^2(mtr)+Sum of {GD^2(i)x[Ni/Nm]^2},

as (i) increases from i=1 to i=n, where the term [Ni/Nm}]^2 represents the individual "inertia ratio's", i.e., the ratio of the (i)th element's
speed to the motor's speed, squared.

Most Common Error
The most common error I have encountered in my experience is the conversion from Metric to English units and vice versa. One vendor supplied a 600 Hp motor, when a 60 Hp was required. The error was two fold:

A) They overlooked the fact that 'D' in the metric equation, GD^2 (kg-mt^2), is the diameter of gyration, while 'R' in the English equation, WR^2 (lb-ft^2), is the radius of rotation.

B) They neglected to square the speed ratio.

The reasons listed above should illustrate the importance of moment of inertia and the inertia ratio. It is a major concern for large motors
because the electrical losses in the motor's rotor during run-up can be significant. It often is the most salient parameter when determining the
number of consecutive starts permitted in one hour. Of course, if rotating mass is negligible, as in small servo motors, then moment of inertia can usually be ignored.

I can supply additional detail, if anyone is interested.

Regards,
Phil Corso, PE
Trip-A-Larm Corp

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#### George Kaufman

Even if the load inertia is ideally coupled to the motor inertia such that the total inertia is the sum of the motor and load inertia, you'll notice that the total inertia inversely lowers the velocity loop gain. From a service standpoint it is good to have the load inertia equal to or less than the motor inertia so that the inertia of the motor primarily determines the
stability. The servo will be stable when the motor is attached to the machine and when the motor is removed from the machine (such as when doing troubleshooting). If the load inertia is much larger than the motor inertia then the servo system will most likely be unstable when the motor is operated off the machine. Matching inertias can avoid misdiagnosis of a problem when a service technician faces an unstable servo when the motor is operated off the machine.

However, in most systems, the load inertia is not ideally coupled to the motor inertia. The belt drives, gearboxes, ballscrews, couplings, and other transmission elements introduce compliance, damping, friction, backlash, possible variable inertia, and sometimes even non-linearities. Once again, the more dominant you can make the motor inertia the easier it will be to tune the system for the desired performance. It's better to have the dog wagging the tail then to have the tail wagging the dog.

Finally, the required bandwidth of the velocity loop is important when considering the inertia mismatch. If high bandwidth is needed, then it is
important to have a good stiff connection between the load and motor and to keep the load inertia close in value to the motor inertia (no more than a 5:1 mismatch). If the bandwidth can be low, I have seen successful applications where the load inertia is several hundred or even several
thousand times larger than the motor inertia. The drive will typically need features such as programmable gains that can be very large (to counter the large total inertia value), low pass filtering on the current commands, and even notch filters on the current commands.

The Handbook of AC Servo Systems provides more details on the above (available as a free download from www.MotionOnline.com).

Regards,
George Kaufman

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#### Johan Bengtsson

Wow, a good question, I have been asking myself that one too I can not se any reason either that it would be of any interest to the controller if the inertia is located inside or outside the motor.

I do however have some ideas where this might come from:

1. If the load are not really tight put to the motor you will have a mecanical system where the potors position and the loads position doesn't necesarily be exactly the same and the connection between these will be close to a spring. This will give you mechanical oscillations between the motor and the load.

2. The controller optimisation calculations is based on a model where the motor inertia is known and the load inertia is given as a factor, these calculations might simply bail out totaly when the ratio it too high but a model and optimisation formula based on another assumption
might work a lot better for big ratios.

It is really going to be interesting to study this in the servo simulation we will do in the near future. You did give me some additional thoughts about things to test....

I do definitely look forward to the answers from the rest of the list...

/Johan Bengtsson

----------------------------------------
Box 252, S-281 23 H{ssleholm SWEDEN
Tel: +46 451 49 460, Fax: +46 451 89 833
E-mail: [email protected]
Internet: http://www.pol.se/
----------------------------------------

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#### Simon Martin

Hi Guy,

>From basic mechanics, inertia ratio of 1 implies the greatest power
transfer.

Empirically, inertia mismatch makes system tuning a nightmare. One of the greatest problems with hydraulic systems is that the inertia of the power
unit (i.e. hydraulic ram) is so low with respect the load inertia, that the tuning is usually the hardest part of the job. I have never enjoyed working on a servo system with an inertia mismatch as high as 10:1, they work but it is difficult to get the correct dynamic response.

Think of it this way, any system like this can be represented as two discs each with a given moment of inertia, representing the motor and load
respectively:

----- -----
| | | |
| |---------| |
| a | | | b |
| |---------| |
| | | |
----- -----

Let Ja be the inertia of a and Jb be the inertia of b.

Case 1 - Ja = Jb
================

Release the coupling between a and b and move a by hand. See the dynamic characteristics of the system. Reconnect the coupling, and apart from the
fact that the system now accelerates at half the rate for a given force, the dynamic characterstics of the system are unchanged. Why? Each load (a and b) reacts in the same way to a given force (conservation of rotational
momentum, energy, etc).

Case 2 - Ja >> Jb
=================

The dynamic characteristics of the system are governed by a, as the highest proportion of the energy required to accelerate, decelerate the system is absorbed by a. If we consider a to be the motor and b the load, then this system is highly inefficient, as the majority of the energy required to move the system is absorbed by the power-unit itself and so is unavailable for the system itself.

Case 3 - Ja << Jb
==================

The dynamic characteristics of the system are governed by b. If we consider a to be the power-unit and b the load, then the dynamic characteristics of the system are governed by the load, the power-unit was optimised to handle its own inertia, but is now having to control a system that is way outside its' spec.

My reasoning in case 3 lacks theoretical background, I have an intuitive feel for it but I don't have a degree in mechanical engineering and I have't really studied in this area. Can anyone fill in the theory here (in words, not maths)?

When you add an encoder, brake, etc you are adding additional load to the system. Why this is considered part of the motor and not the load I do not know. Anyone else?

Debian GNU User
Simon Martin
Project Manager
Isys
mailto: [email protected]

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#### Bruce Durdle

Guy H. Looney wrote:
> Are we to assume that
>by increasing the mass of the motor shaft we can effectively
>increase the motor's rotational inertia? If this is the case, is there
>a limit to how much we can increase it or a set of guide-lines
>stipulating how the mass must be attached? ....That being said, why
>can't we simply put a mass on the shaft of a motor (outside the
>casing) to increase its inertia.

Adding mass to a rotating shaft will increase its Moment of Inertia, regardless of how or where it is attached. But you also need to look at things like the balance of the rotating parts - just sticking a lump of metal on will not be acceptable.

>Some have said that it's a question of compliance & rigidity.
>Assume the load could be pressed on the shaft & that the bearing
> If the drive/motor can handle the torque, speed, & regen is the
>inertia ratio important? If so, at what point does it matter?

The total inertia connected to a motor is not generally going to affect the steady-state
torque or current of the motor. However, it will significantly affect the dynamic performance.
If inertia is too high, the motor will be slow in accelerating and decelerating, and speed control
may suffer. In addition, with inertia connected via a drive train, the natural frequency of the
load and train as a torsional spring needs to be considered - hit this frequency with eg your stepping rate and you are in BIG trouble.. Just as the resonant frequency of a spring-mass combination is dependent on mass and spring constant, the resonant frequency of a torsional assembly depends on the MI at the end and the torsional rigidity of the shaft.

Bruce.

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#### Ken Brown

I waited a few days to see the responses roll in on this one. You have touched on a topic that very few people understand and other than "rules
of thumb", very few helpful thoughts have been put forth by anyone in the motion industry. The best thoughts on this I have read come from a
fellow named Curt Wilson with Delta Tau Data Systems. I will let him spell them out in all their mathematical glory in some upcoming Motion
mag article.

or velocity loops when inertia ratios are such that the load inertia is greater than the motor inertia. Issues dealing with inertia relative to
efficiency and power optimization are easily understood and trivial by comparison.

Every newbie to motion control looks at their tuning charts with perfect step response and 30 - 40 Hz system response and says "wow! . . .this is
going to be cake!" . . . the next thing they do is hook the motor up to a high inertia load and all hell breaks loose and they wonder if they
will ever be able to meet the settling time requirements to do the application.

This is not a nice linear problem and other than the aforementioned author, I have not read one shred of logical evidence for why the inertia ratio should matter one hoot. Lots of words, but no real "stability criteria" based on some mathematical model related to inertia.

Then again, there are the "rules of thumb" that are observable. These "rules" have been changing with time, the better the controllers and
drivetrain components get, the higher the acceptable ratio of load to motor inertia.

The best theory that I have seen published deals with compliance of the drive train. Specifically, the compliance between the part of the motor that you are actually applying controlled torque to "the rotor" AND the part of the system that is reading position "the position feedback device" AND the greatest consumer of this torque . . . in this situation, typically "the load".

One common solution to dealing with high load/motor inertia ratios is to put a gearbox between the motor and load. Immediately you have inserted a spring with backlash between the motor and the load. Unless you plan on always running the motor in one direction with an appreciable
friction load, you will pay dearly for this additional mechanical slop. If your feedback device is on the motor, you could arguably say that things are better . . . more stable, however you have not a clue where the load is when accelerating or coming to a stop, the backlash and
torsional windup in the gearbox makes this a great guessing game. Every time you reverse torque, the load will continue merrily on its way to release the torsional wind-up in the gearbox and then remove the backlash and then re-apply the torsional wind-up. And if you have any
overshoot, you have this cycle repeated with each torque reversal. If the feedback is on the load . . . hysteresis and backlash will keep your
gains low and final position hunting will ensue. This is what "deadband" windows and gains are for.

If instead, you are able to directly couple a motor to the load with suitable torque and speed characteristics with only a very stiff
coupling . . . you have reduced the compliance in the system substantially and will fare much better with settling times and tuning issues. Chances are that the motor with these torque speed
characteristics will also be much higher inertia . . . . hmmmm, where does the credit usually go? This is one reason why AC Vector technology
has so much to offer in motion control applications. Ever compared the shaft diameter of an AC motor rated at 60 lb-inches (typical 5HP motor) and a 60 lb-inch servo motor? You will be surprised, especially when you consider TENV AC motors.

Also . . . putting the "extra" inertia directly on the motor shaft in effect increases the motor inertia . . . nothing to do with the part of
the motor that is exposed to the magnetic field, it just needs to be a very tight coupling of inertia between rotor and inertia hub. This is
why you can order motors with inertia hubs installed in front of and behind the magnet stacks on the rotor. The issue is compliance and
torsional stiffness.

One example of how knowledge of this theory has been applied with a relatively high degree of success with a large inertia miss-match is a
contra-torque system designed specifically for high inertia CNC machines. Using a pair of servo motors driving against a common rack gear, the motors and subsequent drivetrains are preloaded against each other with a balanced torque offset. The torque command from the servo PID loop is split to two DACs and offsets applied at this point, all positive commands go to the Forward DAC /CW drive and motor and all
negative commands go to the Reverse DAC /CCW drive and motor. About 30 - 40% of the system torque is used up in the "contra-torque" counter
balance. At zero speed this chews up a bit of current, but the drives are common bussed, so when running at speed, the regen current from the
holdback drive is fed right back into the motoring drive. What is the result??? Torsional wind-up and drivetrain backlash are effectively
eliminated. A 70 ton bridge is positioned accurately to within .0005 inches quickly and accurately. 1/4 G accels can be had with less than .005" of following error. All with an inertia miss-match of greater than 30:1. In this case, the feedback device was mounted directly to
the load in the form of a .5um laser interferometer.

My money is on the theories that deal with drivetrain compliance between the torque producer inertia and torque consumer inertia within the
positioning system. This along with the placement of the feedback device(s) 'position and velocity' - I think - is where the bulk of the
progress will be made in providing quantifiable answers to the question of inertia matching.

Modeling of all these components is ugly at best. Perhaps the final answer will be uniform torsional stiffness, hysteresis and natural
frequency numbers for all commercially available drivetrain components that can be entered directly into the servo loop's autotune algorithm .
. . . I don't see it happening any time soon.

H. Kenneth Brown
Applied Motion Systems, Inc.
http://www.kinemation.com

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#### andy eccles

The real issue here and also the reason that the feedback device (encoder, resolver,etc) is considered part of the motor instead of considered as part of the load is mechanical resonance. Even in the simplest mechanical configuration there is always going to be a mechanical resonance between the motor and the load. The finite stiffness of the coupling between the motor/feedback device and the load is the spring for this resonance. The stiffer that you make the coupling, the higher the frequency of the resonance. The higher that the inertia ratio is the lower the frequency of the resonance.

When the frequency of the mechanical resonance gets down to a frequency where it can cause the velocity (typically) loop closed loop gain to
increase such that the loop goes unstable you run into trouble.

Andy Eccles

#### PhilCorso

In response to Simon Martin's reply no the subject:

I believe that your perception of the role played by inertia in a dynamic analysis is in error.

Consider the case of a motor accelerating a load (including auxiliaries) from an initial speed, N(i), to a final speed, N(f). Then the time, t(a), it takes to accomplish this is expressed by the acceleration-time curve:

[1] t(a) = K x [integral dN/T(e)] as speed, N, varies from N(i) to N(f).

T(e) is the net torque available to accelerate the total load, i.e., the torque developed by the motor minus the torque required by the load
elements:

[3] T(e) = T(m)-T(l)

K is the inertia constant. It is equal to:

[4] K = 2(Pi)J/60g, where J = moment of inertia and g = acceleration constant.

Note that K is a constant. The only variable is the net torque available to accelerate the load, T(e). If speed dependent then the motor and load
speed-torque curves are used. If T(e) is a constant then the time to accelerate is:

[5] ta = [J/308]x[N(f)-N(i)]/[T(e)]

In conclusion acceleration time is the product of a constant (inertia dependant) and a variable (net torque). The individual moments of inertia of the rotating elements (motor, load, encoder, brake, coupling, etc, reflected to motor shaft speed) are additive.

The inertia ratio is a guideline that says the motor's design torque is "probably" capable of driving an assembly of rotating elements whose
equivalent moment of inertia is "X" times that of the motor's.

Regards,
Phil Corso,PE
Trip-A-Larm Corp

#### PhilCorso

Responding to Ken Brown's reply on the subject, I pose the following:

I do not profess to be expert on servo's. Also, the electrical rotational problems I've experienced were generally void of feedback
concerns and at least two orders of magnitude in size greater than those likely to encounter factors like compliance, rigidity, stability, etc.

But, isn't "compliance" the reciprocal to "stiffness?" Then, if stiffness is defined as a system's ability to resist deviations resulting from loading "forces", how is compliance related to inertia?

Regards,
Phil Corso
Trip-A-Larm Corp

D

#### David Kane

For what its worth, I thought I'd try to explain things in different way.

Inertia is the way that power is reflected or absorbed into the load, under ideal conditions.
At a 1:1 match everything is perfect.

A very high inertia load looks like a mechanical short circuit. The "servo system" is attempting to control this mechanical short circuit, but may not do well.

A low inertia load looks like an open circuit. The servo system has no problem with the load, but the truth is this. The power transfer factor is poor.

Servo motors are typically designed to keep the inertia as low as possible, while producing maximum torque for a given motor design. The combination of inertia and torque (and a few other things thrown in) provide bandwidth. The real target, is high bandwidth. It is most common to use inertia matching as the method to get high bandwidth into the servo designed system.

If a brake added to a servo motor IS included as motor inertia, the motor & drive system would have a lower bandwidth than the same motor without the brake. SO put the brake with the load (not the motor) if you want the system to have the bandwidth of a non-braked motor.

In both electronic and mechanical, maximum power transfer occurs when the power producing inertia or impedence matches the load inertia or impedence.

This is true of audio systems, antennas etc. as well as mechanical

Incidently, in hydraulic systems inertia matching is not typically used. Instead this is calculated as the "Natural Frequency of the Cylinder / Mass
System". The end requirement is the same.

David Kane,
Kane Engineering Group Inc.
AIME
CMCS Certified Motion Control Specialist
[email protected]

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#### Bill Sturm

> The question is why does inertia ratio matter? If the motor/drive
> combination can handle the torque, speed, & regen shouldn't that
> be enough? I have an opinion on this topic but am curious as to
> what others think.

There is an article on this web page called: Load to inertia mismatch: unveiling the truth.

I have always wondered about inertia ratio also. I had thought that as long as you have enough torque, then you can disregard inertia. I assumed that a motor with zero inertia would be ideal. I also could not understand why anyone would add mass to a motor to increase the inertia. It seems like a waste of energy. I had always looked more carefully at total inertia

The reason that the motor to load inertia ratio is important is that a a typical motion system is not made up of one solidly coupled mass, but 2 or
more loosely coupled masses. You have motor inertia and load inertia, and they may not always be in phase with each other. Or due to backlash, there is a switch from motor inertia only to combined motor and load inertia. In other words, the load inertia can vary. With a close inertia match, the varying load inertia is not as large of a change as it would be if the load inertia dominated the system.

With a tight, stiff system, such as a direct coupled ballscrew, you can get away with higher load inertia ratios. If you are using a belt drive or a loose gear train, you would need to have a closer match. Another conclusion that can be drawn from this is that a pulley mounted rigidly to the motor shaft could generally be counted as motor inertia, and would improve the
motor to load inertia ratio, instead of making it worse. How many times have you spec'd a small motor pulley to keep the inertia ratio within range? This may not be necessary, if you have enough torque to accelerate the total inertia at the desired rate.

To summarize: For performance, look at total inertia and total torque. For stability and ease of tuning, you must also look the motor to load inertia ratio and drivetrain compliance.

Bill Sturm
Livonia, Michigan

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#### Ken Brown

Phil,

With regard to my comments and relative to the original post, I addressed the situation where feedback devices are used specifically in servo applications for the purpose of Motion Control. By definition, if no feedback device is used, referring to the system as a servo is a
misnomer.

My personal observation with regard to stiffness and compliance is that it is indeed a significant contributor to instability in servo systems of all sizes. This includes servo systems with ratings exceeding 300 hp used in chassis dynamometer applications. With the high level of
instrumentation and drive control involved in these systems, it is possible to quantify the contribution of torsional deflections in each
part of the drivetrain and correlate this compliance "or lack of stiffness" to factors that contribute to position or velocity loop
instability.

Regarding the concept of having a system with two orders of magnitude greater stiffness than those likely to encounter factors like compliance. Nothing is infinitely stiff and when dealing with
structures and practical design / component specifications some compliance will always be present when dealing with a high performance
positioning system. I think that a concept of stiffness or compliance is typically a relative concept, what one persons idea of stiff might be
another's idea of a rubber band, especially when dealing with high bandwidth positioning requirements.

Best Regards,

Ken Brown
Applied Motion Systems, Inc.
http://www.kinemation.com

S

#### Simon Martin

Hi Phil,

Your analysis is correct for a given mechanical system. The idea though is to compare DIFERRENT mechanical systems, where the load is changed, so that we can see the effects of different ratios. Effectively we are analysing a class of problems where the rotational acceleration (alpha) is equal to the torque (tau) times the moment of inertia (J). J is the composite moment of
inertia of the system, comprised by Ja (motor) and Jb (load). In all cases we get:

tau = alpha * J

The idea is to compare when the cases where Ja = Jb; Ja << Jb; Ja >> Jb, i.e. 3 different mechanical systems

Debian GNU User
Simon Martin
Project Manager
Isys
mailto: [email protected]

K

#### Kaufman, George

"Compliance" is the same as "stiffness" in this discussion (not the reciprocal). The motor inertia is connected to the load inertia with a
"spring". The spring is defined with a stiffness coefficient (torque/radian). Compliant and stiff are just relative terms (such as a less stiff spring is compliant). A true measure of the spring is the stiffness coefficient.

Best regards,
George Kaufman
Automation Intelligence
www.MotionOnline.com

#### PhilCorso

Ken,
I'm not questioning your expertise. I am aware (see "aside below) of the impact that the factors you list have on performance. As I understand
it, "stiffness" is that mechanical characteristic of a translational motion system to resist an inherent weakness in the strength of an element. For example, the torsional twisting of a shaft in a rotating system, or the bending moment of a push-rod in a linear system, or cable stretching in a cable/pulley system.

I should have worded my question differently. Is the "stiffness" characteristic accounted for as an adjustment to the system's moment of
inertia? Or as an adjustment to the net torque (force) available acting on the system?

An Aside.
An analogous characteristic, perturbation, is responsible for our ability to see about 53% (I think) of the moon's surface.

Regards,
Phil Corso

#### PhilCorso

Re: George Kaufman's 4-May, 5:05 pm response,

I disagree. Technically Speaking (borrowing IEEE Spectrum jargon) "compliance" in the context of this thread means "to yield," while "stiffness" connotates "to resist." Hence, the reciprocal relationship.

Re: Ken Brown's 3-May, 3:36 pm response,

My statement "... size" referred to system, i.e., hundreds or thousands of Hp... not to stiffness.

Look fellas, I'm a great believer, and practitioner, in dimensional analysis. All I want is for someone to explain the following:

If, using English units, the moment of inertia has the dimension, ft-lb sec^2, and the dimension for torque is ft-lb, then what is the dimension
for "stiffness?" Or "compliance?" Or for that matter any deleterious "factor." If dimensionless, then, are they used to adjust one or the other of the dimensioned variables?

Regards,
Phil Corso, PE
Trip-A-Larm Corp

A

#### Andy Eccles

The units of torsional stiffness are units of torque/angular displacement such as:

ft-lb/degree
or

Torsional stiffness (the main culprit that makes "inertia ratio" such a big deal) has nothing to do with inertia.

Andy Eccles
Pacific Scientific

G

#### George Kaufman

Stiffness has units of torque/angle or force/distance. You can use any units as long as you are consistent.

For example, the stiffness of a servo is measured by applying a load torque or force and seeing how much position movement has occurred. You can do
this directly to the motor shaft or the load that is driven by the motor.

Static stiffness is measured with a constant torque or force applied. A typical servo with PI velocity control and P position control has infinite static stiffness.

Dynamic stiffness is measured with a step change in load torque or force and the amplitude of dynamic position error is recorded. A typical servo as above can have a dynamic stiffness equal to 15,000 lb-in/rad (but this figure is very dependent on the tuning of the servo).

Please download the Handbook of AC Servo Systems from www.MotionOnline.com for a more complete review of the above principles.

Best regards,
George Kaufman
Automation Intelligence