Deriving the elastica
Deriving the elastica
(OP)
Hi.
I would like to derive the classic Bernoulli/Newton elastica curve from first principles.
Bernoulli's/Newton's classic elastica equation is of the form:
1) curvature = dθ/ds
The RHS reforms to the well known elastica equation:
2) dy2/d2x/(1+(dy/dx)2)3/2
But how do I get from 1) to 2)?
Here are the initial steps I took with no success:
1)ds= √dy2+dx2
2)sinθ=dy/√dy2+dx2
3)cosθ=dx/√dy2+dx2
4)1/ρ=dθ/ds
5)dsinθ/ds=cosθ.dθ/ds=cosθ/ρ
If anybody could help me that would be marvelous!
I would like to derive the classic Bernoulli/Newton elastica curve from first principles.
Bernoulli's/Newton's classic elastica equation is of the form:
1) curvature = dθ/ds
The RHS reforms to the well known elastica equation:
2) dy2/d2x/(1+(dy/dx)2)3/2
But how do I get from 1) to 2)?
Here are the initial steps I took with no success:
1)ds= √dy2+dx2
2)sinθ=dy/√dy2+dx2
3)cosθ=dx/√dy2+dx2
4)1/ρ=dθ/ds
5)dsinθ/ds=cosθ.dθ/ds=cosθ/ρ
If anybody could help me that would be marvelous!






RE: Deriving the elastica
RE: Deriving the elastica
BA
RE: Deriving the elastica
dhengr - a brief look through Timoshenko's "Theroy of Elasticity" yielded no derivation of the elastica I am considering.
BAretired - thanks for the picture, but they also seem to jump a few crucial steps in deriving the formula that I cannot replicate.
Any more suggestions?
RE: Deriving the elastica
Thanks everyone.
RE: Deriving the elastica