Optimizing Cylinder Dimensions for Material Buckling: Calculus Approach?

[sippyCUP]

Homework Statement

This isn't that hard but I cannot remember a nice Calculus way of doing it. I'm trying to find the ratio of height to diameter of a cylinder that produces the minimum material buckling (B_m)^2. The problem statement my professor provided states that the minimum is found at H/D=0.924, but my attempt at substitution has shown otherwise.

Homework Equations

The formula for material buckling of a cylinder is (B_g)^2 = (2.405/r)^2 + (pi/h)^2 which I have simplified to (B_g)^2 = 4[(2.405/d)^2 + (3.1416/2h)^2] .

The Attempt at a Solution

I ran a spreadsheet (attached) which varies the ratio H/D from 0.1 to 1.0. Material buckling decreased even beyond H/D=0.924. I fired up Maple but am not that competent with it so I got nowhere. If anyone has any ideas I would be much obliged.

Thanks!

 

[tiny-tim]

I'm trying to find the ratio of height to diameter of a cylinder that produces the minimum material buckling (B_m)^2.
…
The formula for material buckling of a cylinder is (B_g)^2 = (2.405/r)^2 + (pi/h)^2 which I have simplified to (B_g)^2 = 4[(2.405/d)^2 + (3.1416/2h)^2] .

Hi sippyCUP!

I don't get it …

You're trying to minimise a/D² + b/h² …

but what are you fixing? constant D? constant h? constant volume?

 

[sippyCUP]

Hey tiny-tim,

Both h and d are free to vary... but only the ratio of them matters for the final answer. I'm trying to prove that h/d=0.924 produces the smallest (B_g)^2.

I ran a spreadsheet with d=10 constant and h varying from 1 to 10. This varied h/d from 0.1 to 1. However, the quantity (B_g)^2 decreased the entire time. If h/d=0.924 for minimum (B_g)^2, I should expect it to produce a minimum (B_g)^2 at h=9.24 and d=10.

Obviously the formula does not hold d and h in ratio form. Therefore, I could try changing h to a different value and playing the same game.

EDIT: Having looked at the formula again, I realize this won't help since d and h are the denominators of fractions that will decrease as (d,h) ----> infinity. So I don't really see how the minimum can even occur.