Getting to this 4th order O.D.E.

[hushish]

Hi,

Try as I might, I cannot understand how equation (3) with the k⁴ term was derived. In equation (2), w is a function of t. But in the k⁴ term after equation (3), there is a t² term in the denominator. Should it not be a t¹ only? What am I missing? Please help.

Regards,

Steve

 

[bigfooted]

The screenshot is not so clear. are these the equations?
$w = \sigma_x \frac{l}{R}=\frac{\sigma w t}{R} - \frac{Ety}{R^2}$

and the substitution used is
$\frac{d^2y}{dx^2}=\frac{M}{D}$
$D=\frac{Et^2}{12(1-\mu^2)}$
$w=\frac{d^2M}{dx^2}=D\frac{d^4y}{dx^4}$

This leads indeed to a different answer with t instead of $t^2$, but I do not know where the error is. It could be in the answer or in the substitution rule. You might need to check the second moment of area for your specific geometry to see if it is correct.

 

[hushish]

Hi,

I think the copy of the paper is unclear. I think there is a t³ term in D. Sorry for the confusion...

BUT, I could use your help with the final derivation of equation (4)-see attached. I know that the particular solution to the 4th order O.D.E should look something like this:
y(x) = k⁴(Rδ/E)x⁴/24
But, I don't see that in equation (4). What am I missing?

Regards,

Steve