# Inertia Ratio and Inertia Matching

1. May 12, 2010

### FlexGunship

Okay guys, this one isn't for me; it's for my mechanical brethren.

I'm a motion control engineer and I was assigned to a project to control an axis with high-dynamicism using a low-inertia, high-torque motor. The ratio of the motor inertia to the reflected load inertia is greater than 1:400. Now, to me, it's obvious that there is no shaft coupling in existence that is stiff enough to make this work.

I have shown the mechanical group bode plots, demonstrated regions of stability compared to their terrible step-response and oscillating step-responses with regions of instability. I've tried to explain the dynamics of PID-controllers and the closed loop phase margin. But NOTHING is getting through to these mechanical engineers.

How can I explain to them that their design JUST... WON'T... WORK?!

I show them graphs, draw them diagrams, write down equations... but at the end of the day, this is still some how my problem and I'm supposed to make it work. They keep telling me that the motor has plenty of torque (as if that had something to do with the problem). I calculate a needed bandwidth of 100Hz or more with less than 90 degrees of phase lag, but the open-loop response has 90 degrees of lag before 1Hz!

What words can use to show these guys what I mean? What pictures can I draw?!

2. May 13, 2010

### tvavanasd

You are correct, the torque available from the motor will not help.

The concept of inertia mismatch isn't typically taught to mechanical engineers (not listed in any text I have).

However it does apply to many motion control applications (not as relevant with direct drive rotary or linear motor applications).

It is likely that your load will simply not be controllable as your studies have suggested. Typically, for even moderately dynamic applications, you will want the mismatch to be below 10; the closer to unity the better. The higher the mismatch, the harder the drive will have to work to control the load.

Many gear reducer manufacturers suggest limiting mismatch based on the number of starts per hour to minimize shock loading; motors are often upsized for this reason.

Any good servo sizing software will include analysis of mismatch and include guidelines and explanations.

A good analogy is based on a pair of linked masses with a spring (and backlash) between them - extremes work best.

Imagine a strong man pulling a truck with an elastic rope; the truck not moving until the rope has stretched quite a bit. The man will have a great deal of trouble trying to control the truck due to its high inertia, and the elasticity and backlash in the rope. Imagine also that the man can't continuously feel the tension in the rope, that he can only sample it every few seconds and that he has to stop the truck precisely without looking at it. He also has such a short memory that he doesn't remember that he is even pulling a truck or how heavy it is once it starts moving (until it plows him down). At best, he can sample the tension very frequently (still not enough cause his memory is still too short).

More ideally, 2 billiard balls of identical mass attached by a imaginary string and spring will bounce off each other transferring momentum back and forth with little loss.

The phenomenon is I think analogous to maximum power transfer theory, and momentum transfer.

Can I assume that you are using a gear reducer to reduce the mismatch, and that you are aware that they (gearboxes) reduce inertia by the square of the ratio?

Other options - a direct drive motor that eliminates the need for a reducer and allows for a much higher mismatch (because there is no coupling or backlash).

3. May 14, 2010

### FlexGunship

Thanks for taking the time to address my question.

Firstly, yes, I have recommended a few solutions. The first is a 7:1 to 10:1 low-backlash planetary gearbox with load balancing. The second is switching to a worm gear and pre-loading it to prevent backlash. The third is a much larger motor with a larger shaft diameter (the shaft is the weak link in the coupling chain at ~10^4 Nm/rad); high-dynamic motors, by their nature, have low inertia shafts which makes them, generally, more flexible.

Secondly, by my calculations, there's no coupling on Earth that could cut the phase margin low enough to extend the bandwidth past 20Hz or so. The weak link the motor that the mechanical engineers have spec'd out. The problem, as I'm sure you know, is that torsional stiffnesses add like resistors in parallel. Two coupling in series with torsional stiffness of "x" have an equivalent stiffness of "x/2." No amount of adding stiffer couplings can fix the fact that the motor shaft is not stiff enough. Even a direct drive shaft made of an adamantium/unobtainium alloy would not solve the problem.

Thirdly, your example is great, I used a slightly different one with the same basic idea: try to control the location of a very heavy textbook by using a rubber band and wearing a blindfold. Your knowledge of the book's location is severely compromised. No amount of "tuning" your behavior will ever give you adequate control over the textbook's location.

Lastly, with a ratio of 400:1, I cannot imagine a solution that does not involve a gearbox. My problem is not finding a solution. I have one. It's convincing the mechanical engineers that this is a real problem and that I can't fix it by tuning the PI speed loop, or the PV position loop. They still think I'm just a whiny knucklehead.

4. Jun 19, 2011

### servoguy

I have a very unique solution for the inertia mismatch problem. It eliminates the structural resonances as a limitation. The servo bandwidth is then only limited by the drive bandwidth and/or the sensor bandwidth. I can close the loop above the resonant frequencies with good margins. The technique is very robust. I filed a patent application for the technique in Sept 2009. If you are interested, email me at servoengineer at gmail dot com. I have 45 years of experience designing and building servos and have specialized in servos that control flexible loads.

5. Jun 19, 2011

### servoguy

One more comment: My technique allows the servo to control the load even though the motor is connected to the load by a flexible coupling or gearbox.