How Do You Calculate Equivalent Moment of Inertia for Complex Systems?

[lukea125]

Hi Guys, I'm having trouble finding the equivalent moment of inertia for a system. Basically it's a mass attached to a string, which is attached to a shaft. As the mass drops, the string unravels imparting some rotation on the shaft, this shaft rotates a small flywheel, and also rotates a second shaft via a geared system. This second shaft is attached to a second larger flywheel.

I've run some experiments and determined the frictional torque of the system. However, I can't work out how to find the equivalent moment of inertia. So far I have the energy balance equation:

(0.5I1w1^2 + 0.5I2w2^2 + 0.5mv^2) - T = 0.5 Ie w1^2

where w = angular velocity
T = frictional torque
I1, I2, and Ie are the two moments of inertia of the flywheel and the equivalent moment
v is the velocity of the weight as it falls
m is the mass of the weight

I have the radii of the flywheels, but not the masses, and I can replace w2 with an expression for w1 using gear reduction. Any tips on how I can perhaps eliminate some variables? I can't seem to find a way of solving this, without having the masses of the flywheels.

 

[Unrest]

Where do you want to measure it at? The equivalent rotational inertia will be different on each shaft.

It looks a bit unsolvable without knowing the masses. Unless you know the acceleration.

To include the rotational inertia of the mass on the string, treat it as a point mass located at the radius of the drum. I = m * r²

 

[lukea125]

Ah sorry, I want to find it for the shaft that the string is wound around. I can eliminate omega since it's a common term through gear ratios etc. It's just those moments of Inertia for each flywheel that's killing me.

 

[Unrest]

Ah sorry, I want to find it for the shaft that the string is wound around. I can eliminate omega since it's a common term through gear ratios etc. It's just those moments of Inertia for each flywheel that's killing me.

What are the known variables? I still don't see why it isn't unsolvable.

 

[lukea125]

I've worked it out, Thanks for your help. Turns out we needed to estimate the mass from density and it's dimensions. Made it a lot easier haha.