Inegnuity v's Entropy 2 - Observations & Questions

[ME]

It's interesting to see a top-heavy parallelogram switch between a jacobsladder and a scissorjack:
Is it possible to construct a mech. which acts like B in direction C, and in direction D acts like E.

[Fletcher]

That's the thing about the toy page. We can all see what we want to see in all sorts of combinations. Lord knows I've designed hundreds of Prime Mover variations over the years from that page.

Actually seeing one that has potential to work is hard enough, and it is another exponential level of difficulty again to explain the math that might allow it to work as intended, IMO.

[Fletcher]

Why is that math (and physics) so difficult to identify ?

Because it is such a small effect/gain.

Bessler said that with just one cross-bar it could barely turn itself. But with multiple cross-bars it could achieve a reasonable speed. (paraphrased)

So the very small gain was cumulative and very difficult to identify, else we would just crunch the numbers from known physics and out it would fall I suspect.

But it seems to me that something in our current math does not explain one situation completely and this is what Bessler found and exploited.

That is why I question whether WM2D will ever show the same 'gain' should his Prime Mover be discovered. It may be that you just have to build something and see if it works because WM will show no gain, because that would violate CoE and Conservation of Momentums.

[ME]

When there's such a small gain, it's probable that the winning device shouldn't be able work mathematically or will just balance. In that case WM2D isn't much help, unless we use it differently?
Perhaps we don't need to find a very complex mechanism when such a tiny difference is due to the difference between static-friction and kinetic/rotational-friction; and as such possibly keeping/storing potential until it is able to swing to an other out of balance position - while without friction it should just hang; and thus from a mathematical perspective considered a useless thing.

(as probably noticed: just freewheeling/brainstorming here)

[Fletcher]

When there's such a small gain, it's probable that the winning device shouldn't be able work mathematically or will just balance.

In that case WM2D isn't much help, unless we use it differently?

Exactly, and it is why I use it only to design arrangements and test movements etc, i.e. quickly test functionality.

I never doubt that for almost every mechanical situation that can be built in WM that the sim result will completely match real world observations, for the same inputs. It's completely reliable in this way, IMO. It will never violate any Laws, IINM.

There are however some situations when an arrangements behaviour is hard to fathom, anticipate, and crunch, before I even attempt to build a sim mock-up. At least for me ! One of these may be the odd-ball that doesn't always conform. My bet is that WM would still not violate the Laws (assuming I could build it in sim world and that's not a given, because sometimes the constraints error just won't go away no matter what you do) even if a real world build behaved slightly differently to compare against.

One of those areas is how inertia is treated. Obviously we have mass and inertia in a gravity situation. Mass, for all intents and purposes is unchangeable. Therefore that leaves MOI as a candidate, IMO. That is why I was looking at the static test Driver to Lateral masses examples and their relative capped acceleration relationships. N.B. Of course a wheel in motion is dynamic.

I think when I wrap this thread up I'll end it with one of my favourite gravity-motion Prime Mover mechanism designs that looked good to me.

[Fletcher]

P.S. IIRC, Bessler said something to the effect that mathematicians (of his time) would never solve this.

That tells me it was difficult to crunch once the wheel was dynamic.

I may be wrong.

[ME]

I only attempt a build when I'm confident it will teach me something new; I think it's a costly exercise (mainly time). The rest is done with WM2D or programming where I can let a program do the search through the various parameters. If I would only have the slightest glimpse of the possibility of PM, I would build more.

My main question on this is: can a mechanism be overbalanced over one rotation when the wheel is manually rotated very slow (no dynamic factor) , or is the dynamic factor (centrifugal/inertial effect) that hidden dominant factor - and if so how can this wheel self-start. And for a first PM I don't care how fast or smooth it goes; that's an optimization phase.

Looking forward to see your Prime Mover design.

[Fletcher]

Reply to post from Thu Aug 12, 2015 here: http://www.besslerwheel.com/forum/viewt ... 911#135911

(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.

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 ?????????

I hope you will both have a say in explaining to me the inertial factoring of my formula's.

Fletcher,
Your driver acceleration equation is identical to the Atwoods acceleration equation.
The derivation of the Atwoods acceleration equation for 3 hanging masses was done in this post here using free body diagram analysis:
http://www.besslerwheel.com/forum/download.php?id=10709

If you set m2 and m3 to zero, you get a simpler form of the equation as shown in this diagram here:

http://www.besslerwheel.com/forum/downl ... er=user_id

The acceleration equation for one hanging mass is:
a = (g * driver_mass) / (driver_mass + I/r1²)

This equation is basically: a=F/m
Linear acceleration = Linear Force / Linear Mass

• The acceleration is the linear acceleration of the driver mass.
  The linear force on the system is the driver mass x acceleration due to gravity.
  The linear mass of the system is: (the driver mass) plus (the moment of inertia of the Flywheel divided by the radius squared), where the radius is the radius of the driver mass.

Note: when you take the moment of inertia of the flywheel and divide it by the driver radius squared, you get the "linear mass" of the flywheel that is felt by the driver mass during the acceleration.
To equate this to your gearing problem, the flywheel rim mass represents your lateral load mass.

If you take the right hand side of the equation and divide the numerator and the denominator by the driver_mass, you get:
a = g / {1 + [I/(driver_mass * driver_radius²)]}

Your lateral load mass is the flywheel rim mass. It has a "rotational mass" or moment of inertia of "I".
If you take a moment of inertia and divide it by a radius squared, you get a linear mass.
So the denominator of this acceleration equation is looking a lot like:
(lateral_load_mass/driver_mass) + 1)
But we need to do a little more algebra to get the gearing factor in there.
r1 = radius of driver mass
r2 = radius of flywheel rim mass (or lateral load mass in your case).
If you take r1 and multiply it by some scaling factor (or gearing factor), you get r2.
r2 = C * r1
or:
r2² = C² * r1²

almost there ...
The moment of inertia of a rim mass is I = mr²
If your load_mass was a point mass at radius r2, the moment of inertia of your load_mass would be
I = load_mass * (r2)²
or
I = load_mass * C² * r1²

so the denominator term of the acceleration equation would be

{1 + [I/(driver_mass * driver_radius²)]}
={1 + [load_mass * C² * r1²/(driver_mass * r1²)]}
={1 + [load_mass * C² /(driver_mass )]}
={1 + [C² * load_mass /(driver_mass )]}

Which would make my acceleration equation equal to:
a = g / {1 + [C² * load_mass /(driver_mass )]}

Which matches your acceleration equation of:
Driver Acc = 'g' / ( Gearing^2 * Lateral Load mass to Driver mass Ratio + 1 )

OK .. getting back to this analysis for comparison purposes that Wubbly did. I was looking (in sim world) at if there was anything special in terms of inertia that a storksbill might display which is not covered in current physics. Especially if there was a way to get more momentum transfer from such a system where one mass drops vertically (the driver) and pulls two masses (lateral loads) inwards horizontally at the same time.

He used the Atwoods model as a direct comparison (when the formulas were rearranged). And it shows no energy or momentum gains to be had which is predictable and consistent with current physics.

Just after he posted I ran some sims using just the 'V' sections only of a storksbill to test the relative accelerations of a drive mass located at the joint and equal lateral masses at the ends of the 'v'. They showed a huge difference in accelerations between drive mass and lateral loads as you'd expect as the 'V' closed from wide open to closed shut.

This doesn't easily fit the nice analysis of the pulley system I previously showed (with vertical driver mass geared to vertical load) or the Atwoods situation of unchanging radii of actions.

So how to work it out. Well, we all realize that in my pulley example the 'gearing' is set. This is akin to changing the radius of action in an Atwoods experiment.

So my formula's used a gearing^2 relationship to calculate the MOI effect.

N.B. We also know that say a 1kg load hanging from a pulley will support (no torque) a 4kg load mass with 4 x gearing factor. IOW's, it is the same F1D2 = F2 D1 law of levers at work, that a simple balance beam would show.

So when we look at a closing 'V' arrangement where the accelerations are constantly changing thru the arc of closure we don't have predefined and unchanging gearing ratios. Nevertheless the same effect on MOI is apparent. This is because the gearing or radius squared relationships for pulleys and Atwoods applies to the 'V' as well. But instead of squaring the gearing or the radius we square something else.

Well, that something else is the Speed Ratio from the Mechanical Advantage and Speed Ratio equations of laws of levers.

Efficiency = work output / work input x 100%

since work = f x d

it follows that efficiency = ( load x distance load moves ) / ( effort x distance effort moves )

= M.A x 1 / Velocity Ratio

= M.A. / V.R. x 100%.

So gearing and radii are replaced in my formulas by Velocity (Speed) Ratio to cope with variable accelerations found in these situations.

This makes sense in that distance is all important as we see so often.

..................

Conclusions:

1. There is nothing special about 'V' sections and storksbills in terms of inertia mitigation or ability to transfer more momentum than standard physics predicts.

2. A storksbill is simply a force amplifier ONLY, and nothing more or less ! And does not violate Law of Levers and Mechanical Advantage etc.

[Fletcher]

Here you go ME ..

My earlier favourite Prime Mover based on the Toys Page. "The Kreuz & Crab"

I just looked at the Toy Page and applied the thinking I described earlier in another thread and Ockham's razor.

Ockham's razor : The principle states that

Among competing hypotheses that predict equally well, the one with the fewest assumptions should be selected. Other, more complicated solutions may ultimately prove to provide better predictions, but—in the absence of differences in predictive ability—the fewer assumptions that are made, the better.

It is not OU by itself but perhaps in tandem with another (mirrored or reversed etc) it might do something special, if the connectedness principle is invoked ?!

In principle and in the dual form it possibly does do something interesting in that the lower weights do not have to be 'raised in a flash' i.e. the wheel moves forwards and the load weights remain 'pinned' in space so to speak, but take up a new geometric position for the reset.

IOW's there is no lifting (gain of GPE required) cost for the reset of the lower weights. The Crab and Wheel trade inertia (pulse between them) each half turn IIRC.

Also it reminds me of 'fat lazy horses wandering aimlessly' but then it would wouldn't it.

[Fletcher]

Found it. Here's a Dual Crab version (stylized static drawings CW rotation) where the wheel pulses inertia back and forth during reset twice per revolution.

You can see (look at the arrows to show projected physical displacement) that the lower weights don't move far in 'space' therefore don't actually gain GPE to be 'raised in a flash'. Therefore require minimal Work on them.

That's about as far as I ever took any of this.