- #1

swampwiz

- 486

- 44

Actually it seems that Mohr's Circle is not a tool for just stress or moment if area/stress calculation, but in general for a 2-D simple symmetric form eigenproblem

[ K ]{ x } = λ{ x }

[ K ] → 2-D symmetric matrix

but this analysis seems to only be used in stress or moment if area/stress calculation, I will ask it here as a mechanical engineering topic rather than as a linear algebra topic.

Anyway, it seems that the rotation of axes for a static vector (i.e., the vector stays the same, only the coordinate system measuring it changes) has the form of

[ Ω ] =

[ C S ]

[ -S C ]

which is the transpose (and therefore inverse) of the rotation of a vector within a static coordinate system - and it is this minutiae of detail that I think its causing the problem.

So this rotation matrix becomes the eigenvector matrix for the previously mentioned eigenproblem, as a rotation from the principle aces representation (i.e., [ K ] as a diagonal matrix [ λ ] , which would be the eigenvalues.)

[ λ ] = [ Ω ]

^{T}[ K ] [ Ω ]

[ K ] = [ Ω ] [ λ ] [ Ω ]

^{T}

So from any given original [ K], the matrix for rotated axes [ K' ] would be

[ K' ] = [ Ω ] [ K ] [ Ω ]

^{T}

So going through the math, there is

let X = ½ ( k

_{xx}- k

_{yy})

let Y = k

_{xy}

let I = cos( 2 θ ) X + sin( 2 θ ) Y

let J = - sin( 2 θ ) X + cos( 2 θ ) Y

λ

_{AVG}= ½ ( k

_{xx}+ k

_{yy})

k

_{xx}' = λ

_{AVG}+ I

k

_{yy}' = λ

_{AVG}- I

k

_{yy}' = J

Or at least that it what I come up with - and seems to correlate with that referenced @ foundationcoalition.org - http://www.foundationcoalition.org/resources/en/engr214/Chapter5.pdf (page 5 in the document, or 109 per the header.) However at Wikipedia at an article for moments of inertia - http://en.wikipedia.org/wiki/Second_moment_of_area#Axis_rotation_in_the_xy_plane - the rotation has the signs of the

*sin*terms reversed, which would indicate that the rotation matrix should be inverted - even though that article has the proper rotation matrix!

So my question is which if these references is correct? The foundationcoalition.org document (which my analysis seems to be in agreement), or the Wikipedia article?

If the Wikipedia is, as I suspect incorrect, I will change it.