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

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Discussion Overview

The discussion revolves around calculating the equivalent moment of inertia for a complex mechanical system involving a mass, string, shafts, and flywheels. Participants explore the theoretical and practical aspects of determining this equivalent moment of inertia, considering factors such as frictional torque and gear ratios.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant describes a system where a mass attached to a string causes rotation in a shaft and flywheels, presenting an energy balance equation to find the equivalent moment of inertia.
  • Another participant notes that the equivalent rotational inertia will vary depending on the shaft being measured and suggests that knowing the acceleration could help solve the problem.
  • A suggestion is made to treat the mass on the string as a point mass at the radius of the drum to include its rotational inertia.
  • One participant expresses frustration over the difficulty in solving for the moments of inertia of the flywheels without knowing their masses.
  • Another participant questions the known variables and the solvability of the problem, indicating a lack of clarity on the situation.
  • The original poster later reports success in estimating the mass from density and dimensions, which simplified the calculations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to calculate the equivalent moment of inertia initially, with some expressing uncertainty about the solvability of the problem without certain variables. However, the original poster ultimately finds a solution through estimation.

Contextual Notes

The discussion highlights limitations related to missing mass values for the flywheels and the dependence on assumptions about the system's configuration and parameters.

lukea125
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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.
 
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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
 
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.
 
lukea125 said:
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.
 
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.
 

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