Spinning up a flywheel with a motor

[ag123]

i'm simulating a small dc motor that is driven by volt = 1.54v, current = 250 milli amps, so that gives me 0.385 watts supplied to the motor

next i attached a 4mm thick circular acrylic disc as a flywheel. so in this simulation i spin up the acrylic disc as a flywheel
my acrylic disc is 12cm (4.7") diameter.
acrylic density 1.18 g/cm cube
https://www.engineeringtoolbox.com/density-solids-d_1265.html

next i get the relevant equations:
https://en.wikipedia.org/wiki/Flywheel
E = 1/2 . I . w^2 ... (1)
I ~ moment of internia
w ~ angular velocity

https://en.wikipedia.org/wiki/Torque
power = torque . w ... (2)

then from (1):
E / t = 1/2 . I . w^2 / t
power = 1/2 . I . w^2 / t ... (3)
w = sqrt ( 2 . power . t / I )

now i assume power is that coming from the motor which is 'constant'
so in this way i can plot the angular velocity over time

next subst (3) into (2):
1/2 . I . w^2 / t = torque . w
torque = 1/2 . I . w / t
so in this way i can plot the torque over time
the results look 'surprising', apparently the torque falls even as the acrylic disc / flywheel is spun up (this still seemed quite 'ok')
next the angular velocity i converted it to rpm (revolution per minute) increases but is not linear
what is a little astonishing is the disc spinning at 7000 rpm after 60 secs for a 12cm (4.7") x 4mm thick acrylic disc
that motor delivers a minuscule 0.385 watts !

any mistakes in these calcs?

 

[berkeman]

now i assume power is that coming from the motor which is 'constant'

I'm not sure that some of your assumptions about this DC motor are correct starting points. Where did you get the model? This tutorial may be of help...

http://lancet.mit.edu/motors/motors3.html

 

[ag123]

i've seen that
http://lancet.mit.edu/motors/motors3.html

then as google would take me, I've seen other astonishing charts
https://www.orientalmotor.com/stepper-motors/technology/speed-torque-curves-for-stepper-motors.html
http://www.mechanicalengineeringsite.com/how-to-read-the-speed-torque-curve/

the fictional motor is literally real (the ebay equivalent)
https://www.pololu.com/product/1117

i'm actually trying to measure the stall speed of the motor, but the other curvy wandering charts makes me to curious to probe further. this 'flywheel' is a 'dynamometer', sort of.

if the calcs are correct, i can literally measure torque from the motor in this way, by simply spinning up a disc ! ;)

the 'constant power' assumption comes from the notion that the power supplied to the motor is I (current) x V (voltage). if we assume a 100% efficiency. then that 'power' should be transferred to the flywheel. and if these assumptions and calcs are correct. the flywheel should trace out the rpm - time curve or torque - time curve. the thing is I've nothing to measure torque, i can only measure the speed (angular velocity) . but these calcs are kind of surprising in that small amount of power can drive a rather large (about 50 grams) acrylic disc measuring 12cm (4.7") diameter, 4mm thick to 7000 rpm in 1 minute with only 0.385 watts.

 

[256bits]

the 'constant power' assumption comes from the notion that the power supplied to the motor is I (current) x V (voltage). if we assume a 100% efficiency. then that 'power' should be transferred to the flywheel.

Did you check your calculations as being correct by comparing the electrical energy input to that gained by the flywheel after a time frame.
That is a way to determine if you are on the right track.

 

[A.T.]

but these calcs are kind of surprising in that small amount of power can drive a rather large (about 50 grams) acrylic disc measuring 12cm (4.7") diameter, 4mm thick to 7000 rpm in 1 minute with only 0.385 watts.

It is often "surprising" how little energy mechanical stuff needs compared to heating. For example lifting a certain amount water versus heating it up by 1°.

 

[ag123]

Did you check your calculations as being correct by comparing the electrical energy input to that gained by the flywheel after a time frame.
That is a way to determine if you are on the right track.

think so too, a little trouble is that this experiment if realized would turn into a little project to build a flywheel inertia dynamometer, rpm sensors and all. I'm trying to figure out in what other ways can i use it, e.g. to plot the torque - speed (rpm) curves of the motor. so far only these curves are figured out from the flywheel physics, I've not yet figured out how to create the other data points that would form those important torque - speed (rpm) curves for motors. if building this would only verify these curves, it would feel like a one off use for all that effort

 

[Merlin3189]

" but these calcs are kind of surprising in that small amount of power can drive a rather large (about 50 grams) acrylic disc measuring 12cm (4.7") diameter, 4mm thick to 7000 rpm in 1 minute with only 0.385 watts. "
So you supllied 0.385 x 60 = 23.1 J
The block has a mass of 0.05 kg so 23 J of KE could propel the whole block at 30 m/sec. The periphery of your disc is moving at about 44 m/sec, but the rest of the disc is slower. (I'll leave it to you to do the sum using rotational KE.)

"i'm actually trying to measure the stall speed of the motor, " Zero !

"the 'constant power' assumption comes from the notion that the power supplied to the motor is I (current) x V (voltage). if we assume a 100% efficiency. "
!00% is never true, though it might be good at high revs. At low speed, during early acceleration, efficiency increases from zero.
But depending on when you measured the current, you may have under or over estimated the power input.

"i'm simulating a small dc motor that is driven by volt = 1.54v, current = 250 milli amps, so that gives me 0.385 watts supplied to the motor "
It isn't. If the voltage is fixed at 1.54 V, then the current isn't fixed. If the current is fixed at 250 mA, then the voltage will vary and eventually may not be able to sustain the current anyhow.

Since you are doing a simulation, what parameters can you put in for the motor model?

 

[ag123]

i think these curves, especially the 2nd curve are the 'ideal' curves, as it is based solely on the physics of the flywheel. if i run the motor at different levels of power it would trace multiple sets of the 2nd curve.
what i may have ignored is that for a real motor, the higher end of torque (on the left) may be less, i.e. the motor slips (stall). but then that is strange as it would imply that for a heavy flywheel, it may not turn at all even if the flywheel axle is frictionless. this sounds strange and unintuitive, there seem to be something else missing in the model.

but these curves are interesting if they are true, it gives you the torque-speed (rpm) performance of the motor at a particular level of power

 

[Merlin3189]

It's not clear which curve you are talking about? Is this in post #1 or #2?

That said, I can't see how either implies that a heavy flywheel won't move?
There is a high torque at low (or zero) speed, so it will accelerate.
Where it will not accelerate is when it is already moving very fast at the right hand side of the graph and the torque drops to the level of friction.

One question is where you got your post #1 graphs from? If you got them from your calculations, they are wrong. You must go back to your calculations and get rid of your false assumptions of constant input power and constant torque.

 

[ag123]

Since you are doing a simulation, what parameters can you put in for the motor model?

hi all as it seemed there is some interest in the calcs, I've shared my jupyter-notebook
https://www.kaggle.com/ag1235/motor-flywheel

 

[ag123]

Since you are doing a simulation, what parameters can you put in for the motor model?

Did you check your calculations as being correct by comparing the electrical energy input to that gained by the flywheel after a time frame.
That is a way to determine if you are on the right track.

hi all as it seemed there is some interest in the calcs, I've shared my jupyter-notebook
https://www.kaggle.com/ag1235/flywheel

you can install and run your notebooks locally
https://jupyter.org/

 

[Lnewqban]

... this experiment if realized would turn into a little project to build a flywheel inertia dynamometer, rpm sensors and all...

You could compare yours to typical graphics of inertia dynamometers for DC motors.

Copied from
https://en.wikipedia.org/wiki/Dynamometer

"An inertia dyno system provides a fixed inertial mass flywheel and computes the power required to accelerate the flywheel from the starting to the ending RPM.
It calculates the power required to accelerate that fixed and known mass, and uses a computer to record RPM and acceleration rate to calculate torque."

Please, see:
https://www.minipro.com/products/electric-motor-dyno

 

[ag123]

It's not clear which curve you are talking about? Is this in post #1 or #2?

That said, I can't see how either implies that a heavy flywheel won't move?
There is a high torque at low (or zero) speed, so it will accelerate.
Where it will not accelerate is when it is already moving very fast at the right hand side of the graph and the torque drops to the level of friction.

One question is where you got your post #1 graphs from? If you got them from your calculations, they are wrong. You must go back to your calculations and get rid of your false assumptions of constant input power and constant torque.

this is a surprise too, note that the torque is the implied torque on the flywheel alone if that much power is supplied (0.385 W). it would then goto the notion of motor slip and stall, on the notion that the motor would simply stall and slip if it can't supply that much torque.

my thoughts are that then the flywheel would spin up at a lower level of power, i.e. power = torque x angular velocity, in which the 2nd curve (torque v rpm) would shift downwards nearer to the x and y axis

the curves would also look very different from these 'ideal' curves which is just the flywheel alone.
as initially the motor slip but the flywheel turn with less torque, but at some point the motor would 'catch up' as it can supply the necessary torque and the power transmitted to the flywheel would actually go up

i've shared my jupyter notebook
https://www.physicsforums.com/threads/spinning-up-a-flywheel-with-a-motor.986618/post-6319859

 

[ag123]

" but these calcs are kind of surprising in that small amount of power can drive a rather large (about 50 grams) acrylic disc measuring 12cm (4.7") diameter, 4mm thick to 7000 rpm in 1 minute with only 0.385 watts. "
So you supllied 0.385 x 60 = 23.1 J
The block has a mass of 0.05 kg so 23 J of KE could propel the whole block at 30 m/sec. The periphery of your disc is moving at about 44 m/sec, but the rest of the disc is slower. (I'll leave it to you to do the sum using rotational KE.)

thank you for the calcs, next:

w = v / r = 44 / 0.06 = 733.33 rad / s
rpm = w * 60 / 2.pi = 7002 rpm !
so that's real, and it is this little low power 'toy' motor
https://www.pololu.com/product/1117
wow, maybe i should just try to build it :)

but i think there is something missing, i kind of postulated below, back emf in the motor itself

"i'm actually trying to measure the stall speed of the motor, " Zero !

oops i mean to say the stall torque ;)

"the 'constant power' assumption comes from the notion that the power supplied to the motor is I (current) x V (voltage). if we assume a 100% efficiency. "
!00% is never true, though it might be good at high revs. At low speed, during early acceleration, efficiency increases from zero.
But depending on when you measured the current, you may have under or over estimated the power input.

thanks, the calcs are purely that of the disc (flywheel) itself as the characteristics of the motor needs to be determined. in the charts, the torque is literally derived, i.e. power = torque . w
i don't have a means to measure torque, so this is literally the means.

thinking about this, in a sense i think a motor + flywheel can replace the inductor in a buck converter to make a "buck convertor on steroids", in a sense after the flywheel spins up, it can literally do the reverse and supply power into the circuit and power the load, the idea is very interesting, but 23 J feels little, oh but then 23 J can 'blink' a conventional tungsten filament light bulb, and perhaps quite brightly ;)

"i'm simulating a small dc motor that is driven by volt = 1.54v, current = 250 milli amps, so that gives me 0.385 watts supplied to the motor "
It isn't. If the voltage is fixed at 1.54 V, then the current isn't fixed. If the current is fixed at 250 mA, then the voltage will vary and eventually may not be able to sustain the current anyhow.

thanks, that makes sense, but assuming that the disc is attached to the motor and i spin up the disc while the measured voltage ( 1.54v ) and current (250 mA) stays the same the power delivered would be the same during that time. but of course, this is also an assumption, what would be more interesting is when the disc is spinning up, would power actually go up or down? my thoughts are that it may go *down* as back emf would develop in the motor as rpm increases and eventually it reaches a fixed rpm speed. at that point power input V x I = resistance loss + friction loss + back emf 'loss'
so that the power added would keep the disc spinning at that maximum speed.

unfortunately, this is the part i wish to model as well and I'm currently unsure how to model that process. it would make the model more accurate. i.e. even if i can fix V (voltage), while the disc is spinning up actually I (current) would literally reduce.

Since you are doing a simulation, what parameters can you put in for the motor model?

thanks I've shared my jupyter notebook, it is basically that of the flywheel alone. i hope my model of the flywheel calcs is correct. next like mentioned above, i'd need to figure out that part about the back emf contributions from the motor etc. this simple minded 'flywheel' would then become a very useful model
as it can be used to measure motor characteristics torque - rpm curves etc

https://www.kaggle.com/ag1235/flywheel

 

[jbriggs444]

this is a surprise too, note that the torque is the implied torque on the flywheel alone if that much power is supplied (0.385 W). it would then goto the notion of motor slip and stall, on the notion that the motor would simply stall and slip if it can't supply that much torque.

You have failed to understand post #2. DC motors are not 100% efficient. They do not go to infinite torque at zero RPM. They do not slip. They do not stall. They simply provide non-zero torque at zero RPM.

 

[ag123]

i think there seem to be a way to account for the motor back emf
https://en.wikipedia.org/wiki/Motor_constants#Motor_velocity_constant,_back_EMF_constant

Eb = Kw x phi x w
where phi is the flux, w angular velocity, Kw is back emf constant

however, it didn't seem straightforward to incorporate this into the model. what that would mean is that we need to model the reducing power input as a function of rpm. i.e. the V x I power input will actually *reduce* as the disc (flywheel) spins up, since back emf is proportional to the angular velocity. this is the case where V is held constant (e.g. by the power supply), so the current will keep reducing to some minimum level as the rpm increase to some maximum level, the time based curve will change further to the extent that it will level out at some maximum rpm.

but I'm unsure how to put this in the model so that it can be reflected in the curves.

 

[zoki85]

the 'constant power' assumption comes from the notion that the power supplied to the motor is I (current) x V (voltage). if we assume a 100% efficiency.

I don't understand why do you want to simulate electromotor powered by 1.5 V and less than 1W input.
Do you know how low is the efficiency conversion of electrical input to mechanical output for a very small motor (even of brushless DC type) at high RPM? Very low. Much less than 50%

 

[ag123]

small motors do have uses, they are everywhere. in all sorts of appliances, actuators etc.
and this calc isn't limited to small motors. tiny hard disk motors spins the hard disks at easily 10000 rpm, that's where you get your high Mbytes/s of transfer rates

 

[zoki85]

small motors do have uses, they are everywhere. in all sorts of appliances, actuators etc.
and this calc isn't limited to small motors. tiny hard disk motors spins the hard disks at easily 10000 rpm

And hard disc motor efficiency is?

 

[ag123]

And hard disc motor efficiency is?

near 80% or closer there
https://www.nidec.com/en/product/search/category/B101/M102/S100/NCJ-35R-series/
others vary
https://www.nidec.com/en/product/search/category/B101/M102/S100/NCJ-20N-Type-4/
scroll below for the charts

large motors are not necessarily more efficient that small motors, large high current motors in high torque applications may generate a lot of heat, all that heat is wasted and not used for any useful work.

but that isn't the point, the point is small motors are needed where they are needed. and that this analysis need not be limited to small motors. a lot of small motors do not have specs, especially the *cheap* ones (and it so happens that the 'cheap' ones are the useful ones, sometimes economics determine the motor and not the application itself, so you are left to use 'whatever is most readily available and cheap'). the manufacturer mostly tell you the working voltage and the max currents, my guess is to limit heat output that will damage the motor. if you need things like the stall torque and no load rpm, measure them yourself. for what is worth, i may make do with measuring the stall torque, that's the cheaper, simpler way to do it and no need for flywheel etc.

for large and high quality (including the small ones) motors that cost a premium, chances are that the manufacturer will make nice charts for you and provide various numbers to make calcs convenient

this whole flywheel thing is perhaps useful as it is an analytical model of the motor and a disc (flywheel). the model if it is correct becomes the 'motor equation', and you could use that to determine a lot of characteristics of the motor for sizing (torque-rpm) etc.

the ultimate is for a particular voltage, current - you can estimate the relevant torque, rpm.
it would then have use in control applications. but usually, these days most apps simply measure the rpm for control, e.g. servos

 

[zoki85]

large motors are not necessarily more efficient that small motors, large high current motors in high torque applications may generate a lot of heat, all that heat is wasted and not used for any useful work.

They are if the scalling on the same type of motor is applied.

the point is small motors are needed where they are needed. and that this analysis need not be limited to small motors. a lot of small motors do not have specs, especially the *cheap* ones (and it so happens that the 'cheap' ones are the useful ones, sometimes economics determine the motor and not the application itself, so you are left to use 'whatever is most readily available and cheap')

The point is if you want to have more realistic simulation you need to know η(rpm) curve for a given motor.
For smaller motors the influence of the small efficiency figures is bigger than for bigger motors.
If you want to simulate η=100% cases* than ok, but results will be more or less unrealistic.

* impossible, even in principle for induction motors!

 

[ag123]

well, i don't mean to argue, but this exercise is done in the hope to find a way to measure the *unspecified* (small) motors (inertia flywheel dynamometer), so there is no specs for the motor, this exercise is to derive the specs so that perhaps a later experiment can be done to get the missing parameters or confirm the model.

the other thing is this simulation is currently simply a disc (flywheel), the physics of the disc is deterministic, you can calculate it precisely with physics. these charts in the 1st post are not that of the motor, it is of the disc (flywheel). hence the original model is to assume a 'perfect', situation where the motor produce a constant power, and that power spins up the disc. so you get the charts in the 1st post, I've also shared the jupyter notebook for the calcs.

the model of simply the disc is apparently not adequate to simulate a real case of a motor spinning up the disc.
when the rpm increases, back emf increases, the back emf would reduce the motor currents and hence even if voltage is held constant, power (current) *drops* when the motor spins up. And eventually reach a fixed rpm.

so the charts in the first post are still missing something, the model is incomplete.

 

[jbriggs444]

when the rpm increases, back emf increases, the back emf would reduce the motor currents and hence even if voltage is held constant, power (current) *drops* when the motor spins up. And eventually reach a fixed rpm.

As long as input voltage is held constant, input power and current are proportional. But output torque is proportional to current/input power and output power is determined by torque multiplied by rpm.

One should distinguish carefully between input power and output power. As above, the relationship is via rotation rate.

 

[ag123]

As long as input voltage is held constant, input power and current are proportional. But output torque is proportional to current/input power and output power is determined by torque multiplied by rpm.

One should distinguish carefully between input power and output power. As above, the relationship is via rotation rate.

mostly correct, but the chart in the 1st post show that the output torque may literally behave differently in the case of spinning up a disc (flywheel). Initially i start with a fixed output power assumption, i.e. that the motor is able to supply a fixed amount of output power to the disc. it turns out based on the analysis of the *disc* (flywheel) the torque varies initially high and reduce as rpm increases. And this is not even including the motor in consideration, just disc, no motor.

then i realize something else is missing, when the motor spins up, there would be back emf and back emf is proportional to the rpm. this means due to the rising back emf: if voltage is held constant, current would drop and hence output power would drop. that would be the real scenario, but I'm not sure how to put that real scenario in the chart.

 

[russ_watters]

well, i don't mean to argue, but this exercise is done in the hope to find a way to measure the *unspecified* (small) motors (inertia flywheel dynamometer), so there is no specs for the motor, this exercise is to derive the specs so that perhaps a later experiment can be done to get the missing parameters or confirm the model.

the other thing is this simulation is currently simply a disc (flywheel), the physics of the disc is deterministic, you can calculate it precisely with physics. these charts in the 1st post are not that of the motor, it is of the disc (flywheel). hence the original model is to assume a 'perfect', situation where the motor produce a constant power, and that power spins up the disc. so you get the charts in the 1st post, I've also shared the jupyter notebook for the calcs.

the model of simply the disc is apparently not adequate to simulate a real case of a motor spinning up the disc.
when the rpm increases, back emf increases, the back emf would reduce the motor currents and hence even if voltage is held constant, power (current) *drops* when the motor spins up. And eventually reach a fixed rpm.

so the charts in the first post are still missing something, the model is incomplete.

mostly correct, but the chart in the 1st post show that the output torque may literally behave differently in the case of spinning up a disc (flywheel). Initially i start with a fixed output power assumption, i.e. that the motor is able to supply a fixed amount of output power to the disc. it turns out based on the analysis of the *disc* (flywheel) the torque varies initially high and reduce as rpm increases. And this is not even including the motor in consideration, just disc, no motor.

I don't understand why you are so insistent on using the known wrong assumption of constant power, when the torque curve can easily be programmed into the model. If your goal is to simulate a real motor spinning-up a real disk, you should be using the real performance of a real motor as the basis of the model. Guessing as a starting point isn't very useful, but even worse is sticking to the guess once you find out it was wrong.

You have little hope of constructing an accurate model if you don't change your approach. Can you clarify why you don't want to change your approach?

 

[ag123]

constant power is a simplifying assumption, and that is the initial model. the curves a consistent with spinning up the disc, the disc alone. The motor isn't considered, but the curves of reducing torque as rpm increase make sense, it is not really intuitive as the relationship is non-linear as the curves show in 1st post.

to make the model accurate so that it plots torque - rpm as would be a real motor. i'd need to consider things like the back emf. back emf increases as rpm increases, which in turn reduce currents and it in turns reduce output power so this void the constant power assumption. and it should reach a point the disc simply spin at a fixed rpm.

i'd still think about it but the relationships between the intervening factors backemf, currents, output power again is a rather complicated, possibly non-linear relationship.

 

[cjl]

At a constant voltage, a real motor will not provide constant power (nor will it pull constant current). It will pull peak current near 0 RPM, provide max power at some intermediate RPM, and pull very little current (and provide very little power) at high RPM. Constant power is not a simplifying assumption that makes sense in a scenario like this, and any results generated with that assumption don't seem intuitive because they are wrong.

You can relatively easily find power and torque vs RPM plots for actual motors driven at constant voltage, and you can plug those in parametrically to your model. You don't need to model back EMF or anything, because that's already included in that data. Then you'll get a much more accurate result.

 

[ag123]

thanks all, would try those suggestions ;)

 

[russ_watters]

constant power is a simplifying assumption,

Yeah, I get that, but it is a really bad simplifying assumption, so you should discard it if you want your model to be anywhere close to an accurate reflection of reality.

...and that is the initial model. the curves a consistent with spinning up the disc, the disc alone. The motor isn't considered, but the curves of reducing torque as rpm increase make sense, it is not really intuitive as the relationship is non-linear as the curves show in 1st post.

If I were ignoring the motor, the simplifying assumption I'd make is constant torque. That's easier to incorporate, but it still isn't accurate (note: AC motors are often started and spun-up at constant torque with VFDs). Do we want to model reality here or not?

i'd still think about it but the relationships between the intervening factors backemf, currents, output power again is a rather complicated, possibly non-linear relationship.

The relationships are linear and they really are pretty straightforward:
https://www.micromo.com/technical-library/dc-motor-tutorials/motor-calculations

Your input voltage is fixed and your constants are fixed. So then back-emf is a linear function of rpm. You can plug it all into one equation and solve for torque vs rpm or do it in steps, but the equations and example in that link make it pretty easy.
[note: in the graph, the torque line is mislabeled as "speed"]

 

[russ_watters]

Also, FYI, as far as I can tell, you haven't actually said what you want to use this model for. E.G., once it works, presumably you will do some experiments and apply the model to make predictions. Predictions of what? E.G., what are the inputs you want to measure and outputs you want to calculate? It kind of sounded like you are looking to "measure" moment of inertia...but how?

 

[ag123]

i'm basically wishing to calibrate some generic motors that are available out in the open that don't have specs. if one searches motor literature you would find various charts, these aren't the best ones but it shows the highly non-linear real curves of motor characteristics
https://www.orientalmotor.com/stepper-motors/technology/speed-torque-curves-for-stepper-motors.html
https://commons.wikimedia.org/wiki/File:Torque-Speed_Curve_for_a_typical_AC_motor.jpg

there are attempts to build and measure them like such
https://maker.pro/custom/tutorial/diy-electric-motor-dynamometer-design-principle-and-physics

and as the attempt so far suggest, the 'easy' part turns out to be working the calcs for the flywheel.
moment of inertia is calculated, it is there in the jupyter notebook. the calcs are there
https://www.kaggle.com/ag1235/motor-flywheel

the charts for the flywheel torque and rpm gives a good feel of it but is incomplete as it did not include the motor characteristics in the model. as i work it so far, the conclusions are that it is necessary to account for motor resistive losses and the back emf (which depends on both current and rpm) to arrive at a real model that matches the physical spinning up of the flywheel by the motor at constant voltage.
if the model is incorrect, it is meaningless to talk about using a flywheel as a dynamometer to measure the real motor characteristics. the physics is the basis for that. unfortunately, as i work the calcs, it turns out that those variables resistive losses, mechanical losses, back emf are possibly non-linear and the relationship isn't straight forward to built-in like the flywheel equations.

for now i'd make do with those 'simple' methods that simply measure the stall torque and no load rpm and simply draw a straight line connecting them. those are 'textbook' methods, which are approximations which i'd guess are widely used. the real models that matches physical realities as it seemed are anything but simple (non-linear, has many parameters to account for (resistive losses and back emf are main ones, non-linear current reducing as the flywheel gain rpm, hence reducing power output as rpm increase etc) for something as simple as spinning up a disc (flywheel).

if the model is accurate, i'd be able to model the voltage, current to flywheel torque, rpm like a 'motor equation'

 

[ag123]

thanks russ, your tips helped !

i made some improvements in the model

back emf = k . phi . w = k . i . w
phi is the magnetic flux which is proportional to current, hence substituting current in place
w ~ angular velocity
k - back emf constant
https://en.wikipedia.org/wiki/Motor_constants#Motor_velocity_constant,_back_EMF_constant
this back emf term also accounts for the power output supplied to the disc
https://en.wikipedia.org/wiki/Lenz's_law

voltage is held constant from the power supply in this analysis
voltage = voltage over resistance + back emf
v = i . R + k . i . w
i = v / ( R + k . w )
so in this way current now decreases as angular velocity increases
power output to the disc = back emf x current
Pout = k . i . w . i = k . i^2 . w
now this power is supplied to the disc and spins the disc up

the jupyter notebook is updated
https://www.kaggle.com/ag1235/flywheel

and the charts looks as follows

they still looks quite similar to the 1st post, but there is a surge in torque and power initially as i use the power supplied to the disc as Vin^2 / R only at startup , where R is the motor resistance. this is needed as at start w (angular velocity) is zero and it would give a zero power with the back emf formula (p = k . i^2 . w)

what is noticeable in this simulation is that the power remains rather level (nearly constant), even though current actually decreases. this in part as the increasing rpm actually offset and perk up the output power. the upwards concave torque v rpm chart basically results from a near constant power input as p = torque x angular velocity. Hence, torque = p / angular velocity

efficiency at around 50%, i used Pout / Ptotal = Pemf / (i.r + Pemf) as the formula. resistance value of 1 ohm is used. zoki85 is right, the efficiency of these motors can be low, in particular if the winding resistance is high.

this simulation is probably still imperfect but at least it accounts for the resistance and back emf in some ways. the charts still looked relatively similar as the 1st post

i wondered for a while if using Pemf (the power as is suggested by the back emf) for the power supplied to the disc being correct? i think it is correct, the power isn't lost, the energy is stored in the disc (flywheel) spinning at those rpms. this is probably a simple flywheel energy storage system, spin up the motor, that power is used to spin up the disc. when you cut off power and supposing there is another load (say a filament bulb) in parallel to the motor, the decelerating flywheel and the motor become a generator (dynamo) and transfer that energy out to the load (e.g. filament bulb, keeping it lighted up a little longer)

 

[ag123]

i found another mistake in my model, if i simply use 6 ohm as the motor winding resistance, in the model, rpm reaches 1000 rpm max, and the model is still not 'perfect' all that 1000 rpm is due to the assumption of Vin^2/R being transferred to the disc. i'd still need to fix the model. But the remaining parts for the back emf spinup looks moderately ok.

that 6 ohm probably includes the mechanical friction losses as this model currently only have the resistance as loss. the current motor at hand can reach a no load rpm of 9000 rpm. sounds good.
But the moment a high inertia load such as a flywheel is there, mechanical losses likely shoot up.
no load is simply too light compared to even a moderate small disc.

-- update
for starting up the motor, i assume that the motor is an inductor, the back emf is immediate
but current rises in the 1st second. i calculate that as the energy stored in the inductor (motor)
and use that as the work (energy) transferred to the disc. not perfect, but somewhat better.

#initial starting current,
# assume that motor is an inductor
# emf: v = L . di/dt = L.i.w
# V = iR + k.i.w
# compare with back emf term so L = K
# energy stored in inductor e0 = 1/2.LI^2
i0 = v / R
e0 = 1/2 * K * i0 * i0

the curves are dramatically different

efficiency rise from zero efficiency (at 0 rpm) to 50% (high rpm), seem to make sense. when the motor starts there is little back emf as rpm = 0. the back emf term = k.i.w, i is current, w is angular velocity, k is literally the inductance. coil resistance 1 ohm is used in the simulation

the same jupyter notebook
https://www.kaggle.com/ag1235/flywheel

 

[ag123]

if i use 6 ohm with the inductor starting model, the damage is drastic

60 rpm ! no more 7000 rpm lol
efficiency? zero
this isn't updated on line, it is just a sensitivity analysis

 

[jbriggs444]

If you are getting graphs that are completely out of whack, it's time to stop looking at graphs and start looking at equations. Find a number that does not make sense and track it back to see where that error came from. It will either come from a formula that is incorrect or from another number that is incorrect.

Certainly none of us can help you if you refuse to show your work and, instead, show known bad finished results.