# Stress Analysis for Flywheel

1. Jul 6, 2015

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⋅δrsin(1/2⋅δθ) + σrδθ - (σr + δσr) (r + σr )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.

2. Jul 6, 2015

### Staff: Mentor

This all comes from a differential force balance in the radial direction. The free body has sides rdθ and dr. The σt term comes from the hoop stress. The σr 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

3. Jul 7, 2015

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 σt. I know it's probably just some basic mathematics here, but my experience with limits was a while back and it mostly involved ratios.

4. Jul 7, 2015

### Staff: Mentor

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

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

5. Jul 9, 2015