Gearing MoIs

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Gregory
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Gearing MoIs

Post by Gregory »

The other day I run into an interesting problem:
When a system contains more than one rotating components on different axes connected by a gear train, then how can I represent all the system MoI at one or another end of the gear train?

I found a few pages related to this topic:
http://www.tech.plymouth.ac.uk/sme/desn ... raccel.htm
http://machinedesign.com/sensors/basics ... rol-part-1

As I understand the formula to represent MoI through a gear pair is as follows:

I total = Ia + N^2 * Ib
Ia: MoI of input shaft (object choosen for reference)
Ib: MoI of object geared to Ia (load)
N: Speed ratio of gearing. N= nb/na (number of gear cogs of gears b and a divided)

Can somebody confirm that this is actually correct?
Can this formula be used the same way for epicyclic/planetary gears?

Also I attached a pdf file I found online.
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inertiaformulas.pdf
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Tarsier79
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re: Gearing MoIs

Post by Tarsier79 »

Hi Gregory

I have investigated this before. I thought by gearing something down I could get an imbalance between MOI and RKE.
I spent a lot of time winding fishing line with a weight around pulleys for testing.

The formula you have shown looks right. The formulas for MOI appear to be extrapolated from/ or relate directly to the Kinetic energy formulas. This connection also makes sense if you think about what MOI actually is.
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re: Gearing MoIs

Post by ME »

The formula is correct.
When a system contains more than one rotating components on different axes connected by a gear train, then how can I represent all the system MoI at one or another end of the gear train?
A similar question would be looking at Force and find the representative radius (a gear is basically a neat lever).

Torque = Radius x Force = MoI * angular acceleration
Angular acceleration = Linear acceleration / Radius

You could see MoI as a pre-calculated value for the average Radius of some shape and Mass.
Can somebody confirm that this is actually correct?
Let's say that N=1 then the (ideal) system would be like as if it's mass was doubled and equals the situation when they rotated on the same axle. This hints to the correctness of I[total] = Ia + Ib

With the other N you actually change the leverage as like you would change a radius of a simple lever (Radius*N).
Hence it effects in an effective factor of MoI for Ib:
Ib[effective]= MoIShape.b * Mass.b * (Radius.b*N)^2 = Ib * N^2
Can this formula be used the same way for epicyclic/planetary gears?
It's important to known you drive gear-a, watch the ang.acceleration of gear-b and the gear-ratio N=nb/na.
When you'd drive gear-b and look at the ang.acceleration of gear-a, but keep your N-factor then the total effective MoI would become:
I total = Ia/(N^2) + Ib
This demonstrates the importance of "effective".
But it would work equally well for (ideal) planetary gears, but check the possible increase in contact-friction (more teeth in contact).
Marchello E.
-- May the force lift you up. In case it doesn't, try something else.---
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Re: re: Gearing MoIs

Post by Gregory »

Tarsier79 wrote:Hi Gregory

I have investigated this before. I thought by gearing something down I could get an imbalance between MOI and RKE.
I spent a lot of time winding fishing line with a weight around pulleys for testing.

The formula you have shown looks right. The formulas for MOI appear to be extrapolated from/ or relate directly to the Kinetic energy formulas. This connection also makes sense if you think about what MOI actually is.
Hi Trasier,

Thanks! The reason I asked is not looking for any imbalance. I just have a kind of wheel design and I simulating a key interaction by using a VBA code I have written. It's working fine, but I had to introduce some pseudo variables to get the results I am looking for. Then I realized it is because I forgot to code/represent certain parts of the mechanims/interaction. So, now I will going to do it right as I should have in the first place...
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Re: re: Gearing MoIs

Post by Gregory »

Hi ME,
ME wrote:The formula is correct.
Great!
Can this formula be used the same way for epicyclic/planetary gears?
It's important to known you drive gear-a, watch the ang.acceleration of gear-b and the gear-ratio N=nb/na.
When you'd drive gear-b and look at the ang.acceleration of gear-a, but keep your N-factor then the total effective MoI would become:
I total = Ia/(N^2) + Ib
This demonstrates the importance of "effective".
But it would work equally well for (ideal) planetary gears, but check the possible increase in contact-friction (more teeth in contact).
Right, if we drive gear-b, you wrote:
I total = Ia/(N^2) + Ib

And that equals:
I total = Ib + 1/(N^2)*Ia

And if we would change N to N=na/nb, then from gear-b's perspective we would get:
I total = Ib + N^2 * Ia

Ok, then I think I got it right.
Thank you for the confirmation and analysis. :)
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Post by Gregory »

Another thing I needed to work out for my VBA code is to calculate the velocity of a piston based on only the angular velocity of a crankshaft with r. I figured that a rotating crankshaft with the changing y position of its connecting point to the piston can be thought of as it's being a variable radius pulley where Sin(a) of the connecting point would be the changing radius of the imaginary pulley.

And then the linear velocity of the piston would be: W * Sin(a)
Where W is the angular velocity of the crankshaft, and "a" is the angle of the crankshaft compared to the horizontal line. If I have the inputs in RPM, I only need to multiply it by 2*Pi/60.

But then I concluded that this is not correct for a piston, but this is describing the linear velocity of a Scotch Yoke's sliding part.

I have seen some complicated formulas for the piston, and also derivating with respect to t and such, but I don't prefer it that way, I just want to keep it simple if possible...

I wonder, is it possible to describe the piston's velocity based on the rotation of the crank as something like this:
W * Sin(a) * p;

Where p is some kind of multiplying parameter taking into account the lenght and angle of the connecting rod, etc.
Is it possible to come up with something like that?

Even a good approximation would be enough for me, as well I can use the simple W*sin(a) formula, I was just interested about...
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Post by ME »

If I understand this situation correctly then the easiest ways you could do this VB is calculating the positions (sin,cos) and piston displacement (Pythagoras) and the next one with a temporarily small [dt] to calculate the difference in piston-displacement[ds] and then derive the velocity as v=ds/dt.
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Post by Gregory »

I managed to do what I wanted by using the angle domain equations found here:
https://en.wikipedia.org/wiki/Piston_mo ... _Domain.29

I had to work a little bit on that, but everything is fine, my code works as planned. Except that I encountered another interesting problem as follows:

If I have a wheel with a crank, and a connecting rod which connects the crank to a piston sliding in a straight path, but also sliding inside a perpendicular slotted rod, similar to a simple quick return mechanism for example... Then what will be the "gear ratio" of this connection?
Of course it's not a gear pair, so this might sounds crazy, but still... :D

I thought that the gear ratio of such connection can be represented as the continously changing speed ratio between the crank and the piston/slotted rod. But if I want to represent all the system MoI at the oscillating slotted rod, this causes a "glitch" as the gear ratio converges to infinity (zero speed of piston) at the TDC and BDC point of the piston, which is not so useful in MoI calculations and leading to crappy results...

A possible solution is to coming up with an average speed ratio...
Or, I theorized that instead of the speed ratio, in this case I should really use the radius ratio of the crank and the slotted rod to get a reasonable result with the MoI calculations. Like I can think about them as being "a pair of special pulleys" for simplicity...

Is this assumption correct? And anyway, what is the "gear ratio" of a connecting rod?

Sorry if this is getting messy. I usually solve these conundrums all by myself after some loops of hard thinking, and by any means I will do it again as usual, but any help or insight is appreciated.
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re: Gearing MoIs

Post by Tarsier79 »

I beleive you could calculate your gear ratio using trig, the same way you would a normal lever rotating through 360.
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Post by Gregory »

Yes, I can. I already did it. Sin, Cos, Pythagoras, whatever.

My question was more related to: What gear ratio means for a mechanism without gears?
It can be either speed ratio or radius ratio. I suspect the radius ratio might be the correct one... And I need to calculate an average value (for one full cycle of the mechanism), otherwise it screws up the effective MoI calculations.
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re: Gearing MoIs

Post by ME »

What gear ratio means for a mechanism without gears?
Nothing.

As Tarsier likely hinted: "gear-ratio" is a rotational variant (or sub set) of leverage. And leverage is about mechanical advantage: Force versus distance.
As 'leverage-advantage' (increased force) compresses distance, it automatically effects in reduced speed. The ratio in speed equals the ratio in distance but is the inverse of the Force-ratio...

MoI.
Because each mass-particles at some radius adds to the MoI, you could calculate the average by summing all parts in one rotation, or summing all in partial rotations.
Hence it usually doesn't matter when investigating torque-effects on (let's say) 8 mechanism for 1/8th of a rotation, or just 1 mechanism for a whole rotation, or average 8 mechanisms for one rotation.
...otherwise it screws up the effective MoI calculations
When MoI is some pre-calculated average leverage-effect for mass at some radius it's possible your "effective MoI" may become a useless value when some mass is doing something else besides angular stuff around some axis?
Marchello E.
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re: Gearing MoIs

Post by Tarsier79 »

I don't think you can accurately average, Especially since as you said, it varies between so little, and so much. If you put the formula in a table vs degrees.....

If it simplifies it, you can change your mechanism slightly so the leverage always pushes at 90 degrees to your piston.
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Post by Gregory »

I don't think you can accurately average, Especially since as you said, it varies between so little, and so much. If you put the formula in a table vs degrees.....

If it simplifies it, you can change your mechanism slightly so the leverage always pushes at 90 degrees to your piston.
Looks like it won't be totally accurate ever this way...
But I managed to come up with a kind of average which produces good looking results. I took an average radius(per one cycle) of the lever in which the piston is sliding up/down... and divided it by the piston's crank radius, but without * sin(a), so it stays constant and never gets to 0. Got a ratio of 6.4 for a 0.7-0.58 m lever and 0.1 m crank. I think this is reasonable from one point of view: The connection point on the crank is a fix point, only its circular orientation changes. I think for my purpose this is more or less acceptable. The sin(a) term only matters much if I put force/torque on the piston/crank...
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Post by Gregory »

Nothing.

As Tarsier likely hinted: "gear-ratio" is a rotational variant (or sub set) of leverage. And leverage is about mechanical advantage: Force versus distance.
As 'leverage-advantage' (increased force) compresses distance, it automatically effects in reduced speed. The ratio in speed equals the ratio in distance but is the inverse of the Force-ratio...
Lol, right... I am not trying to use gravity and leverage (or forces with leverage in the usual way) for so long I forgot to think about it like this. But actually I am leveraging MoI against another MoI, so resistance against resistance and not standard forces. When the system is just coasting and there are no outside forces acting on any part of it, plus it is always balanced in gravity field, then what...
When MoI is some pre-calculated average leverage-effect for mass at some radius it's possible your "effective MoI" may become a useless value when some mass is doing something else besides angular stuff around some axis?
I have the MoIs for all the components pre-calculated, and I only work with angular motions. Well, I also have a piston/straight line motion, but that is just a connecting point, it's negligible in terms of MoI.

So, my guess at the radius ratio was in the right direction, aka leverage...

Oh gosh, I have written a code which based on my initial criteria can loop through billions of parameter combinations and find me hundreds of different approximate solutions for an otherwise quite complicated system of integral equations. Then what? I am messing with all this crappy details just to describe an input motion/parameters as accurate as possible. Again I am starting to have that strange feeling that what I am trying to describe is so unusual that there is no standard way to do it properly. I think that's enough for now, I will take an average for that "gear ratio"and see how it works... Perhaps I will try to match my debug/test values against a simple wm2d sim.

Thank you guys!
Good to have your input here as it's forcing me to look on the thing from more different perspectives. If I gonna have any more crazy questions, I'll come back to topic. 8]
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re: Gearing MoIs

Post by rlortie »

Not sure I understand what you seek. If your looking for ratio similar to a internal combustion engine crankshaft, you may find your answer here:

http://spicerparts.com/calculators/rod-ratio-calculator

Ralph
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