🚀 go-pugleaf

Hi Out there,

Heres a problem which has been bugging me for a while now:

Starting off with the undamped biharmonic equation for a vibrating plate:

d^4w + 2d^4w + d^4w = d^2w
----    ----   ----   ----
dx^4  dx^2dy^2 dy^4   dt^2

has the finite element model

        Kw=lambda Mw

where the mass and stiffness matrices contain inertial and stiffness (resp)
terms corresponding to the physical displacement wi, as well as the slopes
and twists at the corners of the elements.

taking Laplace transforms gives

        {s^2*M + K } L(w)= M*x0*s + M*dx0/dt 

Now for a symmetrical square plate, one would expect a certain amount of 
symmetry in w(t) (=L^-1{L(w)), this is not the case.  While the right hand side
of the above equation is a symmetrical vector (symmetrical about its middle
entries) the left hand side is not.

The reason for this is (IMO) the fact that because of the coordinate system
chosen, the slope terms (in the mass and stiffness matrices) on either sides of the plate would be equal in magnitude
but opposite in sign.

viz

	0 0 0 0 0 0 
	0 1 0 0 0 0
	0 0 0 0 0 0
	0 0 0 0 0 0 
	0 1 0 0 0 0
	0 0 0 0 0 0

the elements marked '1' would have the same displacement term, w, the same
dw/dx term (same direction), but the dw/dy terms would be opposite in sign
because of the +ive x and y directions being  to the right and down resp.


Now the questions:

1.	Should only the terms in the mass matrix which involve the displacements	be multiplied by the s**2 terms? (unlikely)

2.	Should I change the coordinate system in some way? (undesirable)

If anybody has any suggestions, I would appreciate an email.

ta muchly,

Martin Crane
Dept. of Comp. Sci.,
Trinity College,
Dublin 2
(400 years young this year)
Ireland