Determining optimum flywheel dimensions

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Discussion Overview

The discussion revolves around the design of a flanged free-spinning flywheel, focusing on determining its dimensions to maximize the moment of inertia while minimizing mass. Participants explore mathematical modeling, iterative calculations, and practical constraints related to the design project on gyroscopes.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Experimental/applied

Main Points Raised

  • One participant presents equations for the moment of inertia and mass of the flywheel, indicating that the only variable is r2, given specific constraints on other dimensions.
  • Another participant shares their approach using Excel for iterative calculations to analyze the relationship between moment of inertia and mass as a function of r2.
  • A suggestion is made to plot the ratio of moment of inertia to mass against r2 to find an optimal point.
  • A participant reports finding a maximum point on the curve at r2 = 42.5 mm and discusses the implications of this finding on spin time, noting variations in results based on different r2 values.
  • Concerns are raised about the impact of bearing resistance and minimum radial load requirements on the design, particularly with respect to mass at the optimal r2.
  • Questions arise regarding the method of subtracting mass from calculations and its relevance to the analysis.

Areas of Agreement / Disagreement

Participants express various approaches and findings, but no consensus is reached on the optimal dimensions or the implications of the calculations. Multiple competing views and methods are presented without resolution.

Contextual Notes

Participants rely on specific assumptions regarding material properties and design constraints, which may influence the outcomes of their calculations. The discussion includes unresolved aspects of the mathematical modeling and practical implications of the design choices.

Who May Find This Useful

Individuals interested in mechanical design, gyroscopic systems, or those working on projects involving flywheel dynamics may find the discussion relevant.

motoxYogi
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Hello all,

I am currently doing a design project on gyroscopes.
My question concerns the flywheel design, specifically trying to determine the dimensions of a flanged free-spinning flywheel to maximize the moment of inertia while trying to minimize the mass, given certain design constraints.
I've treated the flanged flywheel as two separate hollow cylinders and as far as I understand it, the moment of inertia about the axis of rotation is
I=1/2[m1(r12+r22)+m2(r22+r32)]
The mass is
M=πρ[h1(r22-r12)+h2(r32-r22)]
Due to design constraints r3=0.05m and h2 has a maximum size of 0.04m, due to the bearing I will be using r1 = 0.011 m and h1=0.007m. So the only variable is r2.
The material that i have been using is brass, ρ=8400kg/m3

1.png


I know I should be able to use calculus to solve it but I've never been able to apply it well, I've been chasing myself round in circles for days and just keep drawing blanks.
Any help or suggestions is greatly appreciated.
 
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I've been using Excel to perform iterative calculations and then plotted the MOI and mass against r2.
resulting in this graph
2.png

I then tried to approximate where the slope of MOI dropped off significantly compared to the the slope of the mass.
Would this be an acceptable approximation?
 
Plot MOI/Mass vs R2.

Also you can use a subtraction to delete mass (ie lightning it) from the MOI and mass calcs. Assume that all the force acts on the radius of the COM of the deleted segment.
 
Last edited:
Hmmm...
I tried the MOI/mass vs. r2. The maximum point on the curve corresponded to r2 = 42.5 mm.
I then got the bearing resistances for that mass and MOI and plugged it into another excel spreadsheet to determine the maximum spin time it would achieve given initial velocity 1047 rad/sec.
It worked out at approximately 460 seconds. The values I approximated (r2 = 28 mm) worked out at about 550 seconds. When I initially wrote the equations for spin time I was trying to select a bearing, I used a best guess for r2 = 30 mm and it gives me a spin time of 700 sec!
Maybe I should have mentioned the fact that I'm using hybrid bearings, I was just reading through the manufacturers brochure again and realized that it called for a minimum radial load of 16.4 N (1.67kg) for my purposes.
The mass at r2 = 42.5 mm worked out as 1 kg. Would that make a big difference?
Also what did you mean by subtracting the mass?
 

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