Calculate minimum RPM to self-balance a CMG on two legs

[demej00]

TL;DR Summary

I would like to have a formula for determining the minimum RPM a mechanical gyroscope needs to balance itself on two legs.

Here is a photo of a rough drawing of my apparatus that I have built many times and works.

I would like to have a formula to give me the RPM necessary for the gyroscope to balance itself on the two legs (screws). I asked Claude to give me a formula and it gave me the following:

Let me calculate the required RPM foreffective stabilization. I'll use the principles of gyroscopicprecession and the moment of inertia.
First, let's calculate the keyparameters:

1. Moment of inertia of the gyroscopedisk (I):
- Radius (r) = 32.5 cm = 0.325 m
- Mass (m) = 3.5 kg
- For a disk: I = ½mr²
- I = ½ × 3.5 × (0.325)² = 0.185kg⋅m²

2. Required torque to stabilize (τ):
- Bicycle mass (M) = 13.5 kg
- Assume center of mass height (h) ≈0.5 m
- Assume small lean angle (θ) ≈ 5°= 0.087 radians
- τ = Mgh × sin(θ)
- τ = 13.5 × 9.81 × 0.5 ×sin(0.087)
- τ = 0.575 N⋅m

3. Required angular momentum (H) forstable precession:
- For good stability, precession rate(Ω) should be slow, say 0.1 rad/s
- H = τ/Ω
- H = 0.575/0.1 = 5.75 kg⋅m²/s

4. Required angular velocity (ω):
- H = Iω
- ω = H/I
- ω = 5.75/0.185 = 31.08 rad/s

5. Converting to RPM:
- RPM = (ω × 60)/(2π)
- RPM = (31.08 × 60)/(2π)
- RPM ≈ 297 RPM

However, this is the minimum RPM forbasic stabilization. For robust stability accounting for externaldisturbances, you typically want:
- A safety factor of 3-5×
- Therefore, recommended operating RPM:900-1500 RPM


This relatively low RPM requirement isdue to the large diameter and mass of your gyroscope. The largemoment of inertia means you don't need extremely high speeds toachieve the necessary angular momentum.

My question is do you think this is a reasonable answer to my question as to how to calculate the minimum RPM necessary to self balance the CMG on two legs?
My experimental results show that I can balance it at about 600 rpm but only for 10 seconds or so but that could just be due to imbalances in the wheel (I had to balance the bicycle wheel but still out of balance at high rpm) and other friction in the circular bearing, improper CG alignment over the bearing, etc.

I can balance the wheel indefinitely by swinging a counterweight perpendicular to the rotor using two sensors and an Arduino at very low rpm, around 200RPM. But I want to balance it indefinitely without using any active mechanism.

Thanks for any consideration you may give me.

 

[Baluncore]

Welcome to PF.

The speeds seem reasonable.
Why, and for how long, must it balance on two points?

If the points are mounted on the Earth's surface, then the Earth will rotate under it, and it will fall over. Alternatively, it could be operated as a gyrocompass.

 

[Lnewqban]

Welcome!
Are you sure about your bicycle tire and wheel being perfectly balanced and able to remain intact at the recommended 1500 rpm?
That would be equivalent to a bicycle rolling at more than 130 mph.

If the rotating mass is not perfectly balanced, both statically and dynamically, you will always have the undesired influence of a rotating torque working against the stabilization effect.

 

[Baluncore]

My experimental results show that I can balance it at about 600 rpm but only for 10 seconds or so but that could just be due to imbalances in the wheel (I had to balance the bicycle wheel but still out of balance at high rpm) and other friction in the circular bearing, improper CG alignment over the bearing, etc.

You are using the term balance in two different ways. One as the maintenance of a gyroscopic orientation, the other as a dynamic imbalance of a spinning shaft or wheel.

Small round beads, or a liquid, placed inside a spinning tire, will keep the centre of mass of a round wheel, dynamically balanced about the axle. That will automatically reduce vibration of the axle at the frequency of rotation.

Meanwhile, the design of your two-point stand is such that the gyroscope rotor axis, can rotate 90° on the circular bearing, until its axis is parallel with the two points of contact, at which time it must fall over.

 

[demej00]

Welcome!
Are you sure about your bicycle tire and wheel being perfectly balanced and able to remain intact at the recommended 1500 rpm?
That would be equivalent to a bicycle rolling at more than 130 mph.

If the rotating mass is not perfectly balanced, both statically and dynamically, you will always have the undesired influence of a rotating torque working against the stabilization effect.

I have not had it at 1500 rpm, I have had it up to 800 rpm and yeah it was jumping around quite a bit even though I had roughly balanced the wheel.

 

[demej00]

You are using the term balance in two different ways. One as the maintenance of a gyroscopic orientation, the other as a dynamic imbalance of a spinning shaft or wheel.

Small round beads, or a liquid, placed inside a spinning tire, will keep the centre of mass of a round wheel, dynamically balanced about the axle. That will automatically reduce vibration of the axle at the frequency of rotation.

Meanwhile, the design of your two-point stand is such that the gyroscope rotor axis, can rotate 90° on the circular bearing, until its axis is parallel with the two points of contact, at which time it must fall over.

Yes I thought about the beads but decided it was too much trouble to keep going with the bicycle gyro at high speed. So I went the other way and used an active swinging counterweight with two sensors and an Arduino and balanced it indefinitely at 200 rpm.
Yes, it will balance as long as the axis is perpendular to the line of the two legs, at least a small gyroscope will as I have done that with a 3 inch rotor.

 

[demej00]

Welcome to PF.

The speeds seem reasonable.
Why, and for how long, must it balance on two points?

If the points are mounted on the Earth's surface, then the Earth will rotate under it, and it will fall over. Alternatively, it could be operated as a gyrocompass.

Thanks for your response. I have made a 3 inch motor driven rotor gyroscope without any active balancing, balance pretty well indefinitely on two legs. I was hoping to achieve the same thing with my bicycle wheel cmg but it seems the wheel is too unbalanced to spin at the required rpm.

I have no problem getting my gyroscopes to balance on two legs. My real question is does the math look legitimate, as show above, that was provided me by Claude to calculate the required RPM to make a self balancing cmg balance itself on two legs? I don't know much about physics but I can follow these formulas.