Free Vibrations of Plates
A model to be investigated is defined by:
- A polygon set of points that describes the edge of the plate.
- Boundary conditions for each polygon segment. There are three possible specifications:
Displacement w is prevented (blue) or
Incline normal to the edge ∂w/∂n is prevented (green) or
Clamped (cyan).
If only the displacement w is prevented, the support is hinged.
If no specification is given for a segment, the boundary segment is free.
- Specifications for Poisson's ratio ν, density ρ, plate thickness h, and modulus of elasticity E.
Plate Edge
The plate edge is defined using a list of points.
Points can be entered directly or modified using the table.
Additionally, a new point can be created by double-clicking with the mouse.
This new point is normally added to the end of the list.
If there is an empty space in the list of points, the point will be added there.
Points can be moved by clicking and dragging with the mouse.
Pressing Ctrl+B moves all points down one position from the current position, thereby creating an empty space.
Pressing Ctrl+X deletes the point at the current table position.
Plate Vibrations
An elastic, massed plate can exhibit free vibrations perpendicular to its plane of extension, which are investigated here.
For the FEM calculation, the DKT (Discrete Kirchhoff Theory) element according to Batoz is used. It is only suitable for thin plates.
With this FEM triangular element, the natural frequencies and associated mode shapes are determined. A vibration simulation of the plate in its eigenmodes is also performed.
In principle, only certain modes and frequencies are possible for free vibrations of elastic structures, so-called eigenmodes with their associated eigenfrequencies.
However, in an elastic structure with distributed masses and stiffnesses, there are infinitely many such eigenfrequencies.
Since often only a few of these, namely the lowest, eigenfrequencies are of interest, the higher eigenfrequencies can be omitted from the calculation.
Restricting the calculation to a few lower eigenfrequencies is advantageous for several reasons. Firstly, the computational effort increases rapidly when using models with many degrees of freedom.
Furthermore, the eigenfrequencies determined with FEM become increasingly inaccurate with increasing order. The model is generally too rigid for them.
Furthermore, the mode shapes of the higher natural frequencies are more difficult to excite in the real structure. They are also damped more strongly by structural damping.
Defining the respective problem essentially requires two steps.
First, the plate geometry must be defined.
It must be defined by a closed polygon of the boundary points.
Then, the regions along the boundary where the plate has support conditions must be specified:
- w=0, hinged support of the edge, anti-symmetry boundary condition
- ∂w/∂n=0, inclination normal to the edge restrained, symmetry boundary condition
- w=0 and ∂w/∂n=0, edge clamped
If no support conditions are specified, the natural frequencies and mode shapes of the unconstrained plate are determined.
Then there are always three rigid body mode shapes (two rotational and one translational) with a natural frequency of 0.
These are of course not vibration modes, even though they are represented here by motion.
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