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Finite element model of a viscoelastically supported plate

Finite element model of a viscoelastically supported plate

Finite element model of a viscoelastically supported plate

(OP)
Hi,
I currently have the FE model of a plate, that is, the matrices [K] and [M] are already built.
Now I would like to add the effect of viscoelastic supports, located at the plate's edges, to the stiffness matrix [K]. Those supports have a complex stiffness
in the form K(1 + in).
Is there a straightforward method of accomplishing this?
Thank you very much.

RE: Finite element model of a viscoelastically supported plate

You can use spring elements that allow you to input a stress-strain curve (like a non-linear truss element).  Calculate the stress-strain curve in such a way that the modulus of elasticity changes in accordance with your required support stiffness.  Use a cross-sectional area of 1 and a uniform length.  The stiffness can be calculated as AE/L.  If A is '1' and the starting length is L, E is determined from the stress-strain curve that you are inputting...

RE: Finite element model of a viscoelastically supported plate

(OP)
Thank you GBor. I actually have the value of the edges' viscoelastic elements, expressed as a complex stiffness (where the complex component represents the energy dissipation). What I would like to know is how to directly add that stiffness to the stiffness matrix [K], which already exists.
I'm using MatLab to do the programming.
Thanks!

RE: Finite element model of a viscoelastically supported plate

Since this is a time dependent problem (the time dependence coming from the behavior of the VE material of course), typically this is handled with an iterative process, say with a Newton iteration. You can look in the Bathe FEA book for details, but basically it works like this. The principle of virtual work {PVW} (which are the integral equations from which you derive KU=F) is broken up into a couple of parts. If you recall the Newton iteration, the new guess 'x(i+1)' is dependent on the previous guess 'x(i)' plus an increment. Inside the PVW are stresses and strains, and you break those stresses and strains into the new guess 'sigma(i+1)' and the previous guess 'sigma(i)' and an increment. You can then construct the finite element matrices KU=F; with no time dependence or other nonlinearity, "F" contains only the force vectors due to the external loads and/or tractions and terms with the springs (some people like to fold those spring terms into the K). However, with time dependence, "F" now contains not only those terms associated with the tractions, but terms due to a KU(i), that is, the KU at the previous time step.

With your time dependence, you would assume you are loading this structure with a time dependent force--even if you want the force to be P, say, you artifically create a ramp up in force from time=0 to t=tmax, and how fast you get there (that is, how many or few time increments) has to be tinkered with, since you can make this problem numerically unstable if your time increment is too large.

Because of the time dependence of the VE support, this is an intrinsically nonlinear problem, requiring some sort of interative numerical method like the Newton method.

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