bluesgtr44 wrote:It's been awhile since I have put anything up.
. . .
The one factor that surprises me that is not available......is the time. I cannot find any reference as to the time it took for the wheel to reach maximum speed. There are references as to how many revolutions, but nothing about the time.
You asked this before:
http://www.besslerwheel.com/forum/viewt ... 0816#10816
Perhaps I can do some rough guessing...
Let's say the rotation of the unidirectional wheel starts from a fixed position and the Center of Mass lies in the maximum overbalance position; that is: the averaged position for all weights together;
When it accelerates towards maximum velocity (within about 2.5 turns, or 900°) the overbalance keels towards the ground as the mechanism can't keep up with raising some weight.
Perhaps things aren't raised and overbalance is caused in some other mechanical way - which simply takes time until the wheel moves at such a speed it doesn't effect in overbalance anymore.
(sidenote: I don't know if such logic still counts for Jim's motion wheel)
The maximum possible acceleration would be g=9.81 m/s^2, but it should be (much) less because of inertia of the wheel itself; so acceleration a[initial]=f*g, where f is some unknown factor [0<f<1]
As the average overbalanced position keels that acceleration will go to zero, depending on the angle of such overbalance, which depends on velocity;
I think a good guess would become:
a[v] = a[initial]*cos( 0.5*pi*v/v[max]) = f*g*cos( 0.5*pi * v/v[max])
(an acceleration as function of the current velocity, which depends on the lift angle, which depends on its deployment time at the current speed)
-There could be so much wrong with this assumption-
v[t]=a[v[t]]*t
As one can see this is a somewhat circular definition: the acceleration depends on its current velocity (thus also the speed of lifting a weight), and wheel velocity depends on acceleration. Perhaps I should redefine this stuff, but I'm too lazy for that.
It gets slightly more complicated when calculating the initial acceleration to deg/s^2
For all this we need the mass and mass distribution of the wheel to calculate the inertial moment of the wheel;
When we know how much a weight is raised (or some equivalent action) we know the maximum overbalanced position ( h/pi ) horizontally, and we need to know the total amount of weights in CoM - as the weights supposed to work in pairs it is difficult to say what that would do to the CoM of those weights combined, or this should only affect the inertia of the wheel of something in between.
Let's make some rough guesses:
Diameter 5 ft: 0.762 m radius
Just wheel Mass: 40 kg ("easy lift" for two men?)
Mass distribution: Even distribution (probably not)
Inertial moment 0.5*40*0.762^2 = 11.613 kg m2
8 weight-pairs @ 2 kg = 2*8*2 = 32 kg
Max.Rotation: 50 RPM = 300°/sec
Amount of rotations: 2 or 3 --> 2.5*360°=900°
While I use math I'm no math-geek, so I let the computer run the numbers (what else should that thing do).
- - - -
Initial acceleration: a[init]=135.039 °/s2
I have a scenario where a single mechanism lifts 28cm, two times per rotation.
At 900° rotation it reaches a velocity of 285°/sec, which is (my target) 95 % of the maximum velocity.
(In this exercise it will never reach the maximum, but just gets closer)
For the rough calculations I (and my PC) performed, it should do this in 4.576 seconds.
It's likely that only half of the weights are lifted, perhaps each of one pair only lifted once per cycle.
So I did fix half of them at an arbitrary radius of 0.45, and basically they're just increasing wheel inertia.
To get to the 95% max. velocity I had to raise the weights a distance of 0.742 m, basically from rim towards the center of the wheel.
Initial acceleration: a[init]=134.854 °/s2
The time is 4.578 seconds, about equal as before.
This all with the obvious disclaimer my method and/or calculus could be completely or partially wrong; but it's the best I've currently got.