# Inertia ratio

J

#### John Boland

Phil Corso writes:

<< 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?" >>

A good physics or mechanical engineering text should cover this.

May I suggest working with linear motion in MKS units first? The English "system" is filled with derivative, not fundamental, units (pounds, for
example, versus slugs) and rotary motion has those nasty radians floating around.

Dimensional analysis falls apart in our (electrical) world, too, if we treat amps and ohms as fundamental units instead of using:
E (volt)
I (coulomb / sec)
R (volt - sec / coulomb)
Watt (volt - coulomb / sec)

This "Inertia Ratio" issue may result from an over-simplification of the mathematics of the natural frequency of the bare motor and its stable
control, versus the natural frequency of the load. Not torque, speed, or acceleration, ...

We know that a bare motor's internal magnetic / mechanical frequency response can be poor - for example, with a "normal" induction motor and a VFD - or spectacular (to me) - with a DC servo. Rigidly couple equivalent HP / RPM / inertia rated motors of these two types to an equivalent - inertia flywheel and test the frequency responses. There will be a tremendous difference,
although the "Inertia Ratio" is 1:1.

Couple the same DC servo motor to the same equivalent - inertia flywheel with a low "K" torsion spring (where "K" is the torque per resulting radial deflection - in units of your choosing). Test the frequence response. You can
make it as bad as you want by reducing the ratio of spring stiffness to load inertia (softer springs).

In electrical terms, the electro-mechanical system behaves like a lumped transmission line.

Have a good weekend, folks. I got my asbestos undies on.

"Lurker John" G. Boland, president
VisiBit Corporation
One Parker Square Suite 408
2525 Kell Boulevard
Wichita Falls, Texas 76308
940.322.9922
940.723.1478 fax

D

#### David Kane

Bill Sturm wrote..........
> 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

This design method would not work for precision servo control with a dynamic load, because the power transfer could be unacceptable for the machine system.

Example: A point to point transfer can usually get away with more inertia mismatch. This is because there is probably not a "dynamic" disturbance attempting to act on the load.

Consider a Precision CNC Machine Tool. The cutting tool is doing to try to push the the servo(s) out of position, even introducing possible tool chatter. This is a high dynamic load disturbance. As cutting speeds increase tool marks are produced at higher frequencies. A higher bandwidth capability results in more accurate part finishes. A 1:1 match will compensate much better than a 2:1 system.

> 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.

Inertia can change if the attached load is a cam, or crank linkage. However, The inertia is not changing and does not vary as described. Displacement changes or spring as described are a separate 'coupling' term, in control design process. This term reduces frequency response and is disadvantagous to the system.

Assume this mechanical did a poor mechanical design and the mechanical system has a response 2 Hertz, And assume the Inertia is match and happens to provide a 10 Hertz response. Take a guess which frequency will dominate!!!! The machine servo system response will be less than 2 Hertz.

The servo system cannot make the machine respond any faster then the natural frequency of the machine. This corresponds to the design and construction of the machine. Couplings, ballscrew diameters, belts, backlash, etc. all
affect these things.

>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.

This is not correct at all. The pulley on the motor is load inertia. Occasionally adding inertia will move a resonant frequency, making the
system seem to sound or act better, but from a dynamic point of view the response is actually slower. The addition of load inertia will in all cases reduce bandwidth.

Summary.... All machinery has a natural frequency. It is a function of the machine design and construction. Inertia matching provides optimised natural frequency for the servo control system with reference to the machine, but it does not address the machine
natural frequency.

The benefits of a high performance (inertia matched) servo system can be lost if the machine is not designed to accomadate the benefit.

In the case of a simple point to point transfer, it may not be economical to design with zero backlash couplings, larger ballscrews, etc. to acheive a machine with high natural frequency.
If this is true then designing with larger inertia mismatch could be acceptable.

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

#### PhilCorso

In my 4-May-00, 2:53pm response on the subject, I provided an "aside"... a bit of trivia about the Moon. It revealed that due to "perturbation"
we earthlings can see 53% of Moon's surface.

I erred.

Actually, we see about 59%. This observation is due to a phenomenon called "Lunar Librations." The Moon appears to nod up and down and left
to right. For detailed information Search Web for "Inconstant Moon." I don't have its URL.

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