Inegnuity v's Entropy 2 - Observations & Questions

[Fletcher]

Here is the spreadsheet analysis for this Geared Driver and Lateral Load scenario.

I built matrices to see patterns in the numbers, and I created fields for Inputs and Outputs.

The blue numbers are actual sim results for accelerations and forces. The black numbers are my acceleration and force calculations generated by my formulas.

I started by building a formula for the 1 x Gearing option. This was easiest to solve. Then I adjusted the equation elements for the other gearing options until the formula matched the sim in all gearing options and with any mass combinations.

[ME]

Although I have a formula which can calculate the acceleration... I don't understand it (yet?)

add The acceleration of the [Lat.Mass] depends on the gravitational attraction of the [Drive.Mass] but that depends on the ratio, while the mass leverage also depends on it.... (or something like that)

[ME]

... it acts like a pendulum.

[Fletcher]

Although I have a formula which can calculate the acceleration... I don't understand it (yet?)

add The acceleration of the [Lat.Mass] depends on the gravitational attraction of the [Drive.Mass] but that depends on the ratio, while the mass leverage also depends on it.... (or something like that)

Accelerations are fundamental. But they are a function of inertia. From them other things 'downstream' can be derived.

You are correct. If you can calculate the new acceleration of the drive mass then the lateral load acceleration will be a function of the gearing ratio.

[ME]

The funny thing is that the following setup gives the same result, where ratio (n=r.drive/r.lat) and of course only for (t=0).

[Fletcher]

Here is the formula.

There are three alternative arrangements (see pic below).

Basically you need to start with the Driver acceleration first which is 'g' -10 m/s^2.

Then you find the new Driver Acc output by adjusting for inertial factors which includes Lateral Load mass to Driver mass Ratio and Gearing used.

(I believe Gearing is squared in the formula because KE Output for CoE is a squared term).

e.g. for Driver mass 4.0kg and Lateral Load mass of 0.5kg with 4 x Gearing ...

* Driver Acc = 'g' / ( Gearing^2 * Lateral Load mass to Driver mass Ratio + 1 ) => -10 / ( 4^2 x 0.125 + 1 ) = -3.333 m/s^2

The Lateral Load Acc is 4 x the Driver Acc because of the gearing used.

* Lat Load Acc = - Driver Acc x Gearing => -1 x -3.333 x 4 = 13.333 m/s^2

N.B. the Lateral Load acceleration has a multiplying factor of negative 1 because of the direction of the acceleration.

You could also use the whole Driver Acc formula in this next formula.

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Once we understand pulleys then I will explain the significance and possibly potential of the storksbill as I see it which obviously has variable accelerations for one thing.

[ME]

I've written it differently... but it's the same formula

mD=Drive Mass, mL=Load Mass, n=ratio, g=gr.acceleration

aL=g * mD / (mD/n + mL*n)
aD=aD / n

edit: (oops, swapped D and L - I use different designations)

[Fletcher]

Then you find the new Driver Acc output by adjusting for inertial factors which includes Lateral Load mass to Driver mass Ratio and Gearing used.

(I believe Gearing is squared in the formula because KE Output for CoE is a squared term).

Can someone please explain to me why we have to square the Gearing factor when considering the inertia quotient effect on the new acceleration of the Driver ?????????


It is very important that I understand this requirement !


Alternatively you might have a different formula with different elements ?

[Fletcher]

I will explain my concerns and reasoning.

For me to match the sim predictions of Driver Accelerations etc I had to figure out a formula starting with the only force available. That being the Driver weight force ( m x a ) in N's. Then if I removed the mass component of that weight force I was left with 'g', as you'd expect.

Then I had to adjust 'g' acting on the Driver to less than -10 m/s^2 because of the inertia of the Driver and the Lateral Load moving horizontally.

That meant there was a Ratio involved. Then I had to adjust that answer for the Gearing used.

BUT .. for linear motion I = m. So I was a bit surprised to have to square the gearing influence.

For example (using Dr mass 4kg ; Lat Load 0.5kg) then the new Driver Acc with 4 x gearing is -10 / ( 4^2 x 0.125 +1 ) = -3.333 m/s^2.

For 2 x gearing the new Driver Acc is -10 / ( 2^2 x 0.125 + 1 ) = -6.667 m/s^2 (twice as much as 4 x gearing in this example).

BUT .. we know that Rotational Inertia is something like mr^2 and when the radius gets bigger the inertia increases by a squaring factor.

I wasn't expecting this in a linear relationship. I know that it is completely necessary in order for CoE to hold true in sim world i.e. when you calculate the force x displacement values you get Joules, and these sum to the KE's the sim shows.

[Fletcher]

Then on thinking about it I remembered that gearing was the same as a lever and fulcrum arrangement. And that would be subject to I = mr^2.

Then it made more sense to square the gearing influence modified by the ratio of masses.

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The formulas for next two scenarios are even more strange (that was the easy one).

[Fletcher]

ETA: That meant that for both rotational motion and linear pulleys (geared) that the inertia of the masses is a squared function of radius and gearing respectively. This was the surprise i.e. that for pulleys I =/= m.

It seems that the distance moved (and velocity) is the critical factor in arriving at the inertia.

Does anyone have any other ideas ?

[Tarsier79]

I had a similar thought a while ago. I built a test rig that was a geared 2:1 ratio, where I could install the inertial weights at 2x radius or at 1x radius geared to 2x speed, all driven with a falling mass. The results were opposite than expected, but I think it was due to extra friction and load on the chain.

[Fletcher]

Yeah .. seems to confirm that Rotational Inertia is mr^2 if nothing else.

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Here is the formula and pics for the next scenario.

Geared Vertical Driver mass to Vertical Load.

Note how the inertia of the masses in opposition must be modified further with a somewhat bizarre extension to the formula (it took some working out !).


e.g. Dr Mass 4kg ; Load Mass 0.5kg ; 4 x Gearing

Ay Driver m/s^2 = - 'g' / ( G^2 * Ld : Dr Mass Ratio + 1 ) * ( G * Ld - Dr ) / Dr )

Ay Dr = 10 / ( 16 * 0.125 + 1 ) * ( 4 * 0.5 - 4.0 ) / 4.0 ) = -10 / 3 * -0.5 = - 1.667 m/s^2

Ay Load m/s^2 = - Driver Acc * G

Ay Ld = 1.667 * 4 = 6.667 m/s^2

Yes, the formulas work for all mass combinations etc.

I can't really explain the formula extension this time ?! It just works !

[Fletcher]

Here's the third scenario in this series.

Geared Load Mass to Driver Mass.


e.g. Dr Mass 4kg ; Load Mass 0.5kg ; 4 x Gearing

Ay Driver m/s^2 = - Load Acc * G

Ay Dr = - 2.403 * 4 = - 9.612 m/s^2

Ay Load m/s^2 = - 'g' / ( G^2 * Dr : Ld Mass Ratio + 1 ) * ( G * Dr - Ld ) / Ld)

Ay Ld = 10 / ( 16 * 8 + 1 ) * ( 4 * 4 - 0.5 ) / 0.5 ) = 10 / 129 * 31 = 2.403 m/s^2


Note the formula extension looks similar to the last but is not the same.

[Fletcher]

So what do these pulley sims and others teach us ? JMO's.

That it is unlikely that every random sim possibility is pre-programmed into the sim. That would mean an inordinate amount of formulas specific to each situation be encoded.

It is more likely that the sim works by enabling a Top Down approach and assumes CoE, and then runs iterative calculations to find the energies and by reduction the accelerations etc. But then I'm not a programmer of Kinematic systems.

N.B. WM2D has a force element that can be added to a sim. It can cause a sim to increase in energy levels. This represents some external force entering the system, and so, a sim can demonstrate OU when tweaked this way.

That means that a sim will never demonstrate OU without the introduction of an external force to the system. But most of us suspected that anyway and use it for investigating movement of parts and design etc. That is its real strength, and its simplicity and expediency.

Specific to pulley arrangements:

Pulleys demonstrate constant accelerations. These are invariable determined by Gearing and mass relationships.

They show that system KE's will never exceed GPE's lost in a frictionless environment.

IOW's, Net GPE lost equals Sum KE of all system parts. This has not been tested against a physical build for verification but is axiomatic I believe.

i.e. A build in real world (as close to frictionless as possible) would demonstrate similar results to sim world which would however be more efficient.

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What good qualities and potential might Storksbills have in their geometry and action to make them special in any way as a Prime Mover candidate ?