How to Derive Equation 2 for Stress Analysis in Flywheels

[James Brady]

Hello, I'm trying to follow along with the stress analysis derivation for flywheels given here, but I'm stuck at the point where it says:2⋅σ _t⋅δ_rsin(1/2⋅δθ) + σ_rδθ - (σ_r + δσ_r) (r + σ_r )θδθ = ρr²ω²δr ⋅δθ

in the limit reduces to:

σ_t- σ_r - r⋅\frac{dσ_r}{dr}= \rho⋅r^2 ω^2

I'm a little rusty on limits and how to perform them. If you follow the link, there's a pretty good drawing of the differential element which explains equation 1. I'm just not sure how to get equation 2.

 

[Chestermiller]

This all comes from a differential force balance in the radial direction. The free body has sides rdθ and dr. The σ_t term comes from the hoop stress. The σ_r terms come from the radial direction, and takes into account the variation of r across the free body radially. The term on the right hand side is the centripetal force term.

Chet

 

[James Brady]

I understand why all the forces on the stress element are there. I just don't understand how the limit is taken. For instance, why the first term, 2⋅σ t⋅δrsin(1/2⋅δθ), reduces to σ_t. I know it's probably just some basic mathematics here, but my experience with limits was a while back and it mostly involved ratios.

 

[mfb]

$\sin(x) \approx x$ for small x.

All terms have δr δθ as common factor at leading order, which gets removed to give the second equation.

 

[James Brady]

Ah, the small angle approximation. I totally forgot about that. Thank you.