(i) It is desired to develop a finite element for the transverse vibration of a moderately thick...

(i) It is desired
to develop a finite element for the transverse vibration of a moderately thick
plate described by the first-order zig-zag theory. The finite element is
assumed to be rectangular (2a × 2b) in shape with the element coordinate system
centred at the mid-point of the plate. The four nodes are defined in terms of
non-dimensional coordinates at ξ = ±1 and η = ±1. Assume a
three-layer laminate with the middle layer non-piezoelectric. All layers are
assumed to be of the same thickness. The displacement and rotations at each
node are the element degrees of freedom. The displacement is approximated as

and k is the layer number. It is zero for
the middle layer, 1 for the top layer and −1 for the bottom layer. (ii)
Derive the finite element stiffness, mass and forcing vector distribution
matrices when the transverse excitation is for a bimorph configuration. (iii)
Using MATLAB/Simulink, model a typical rectangular plate by 10 × 10 elements
and establish the equations of motion for the plate in a global reference
frame. Obtain the natural frequencies and mode shapes of the plate. (Save your
m-files for use with later exercises.) (iv) Compare your results with the
results obtained in the previous.