(i) A finite beam element for modelling the transverse bending vibration is required to be...

(i) A finite beam element for modelling the
transverse bending vibration is required to be developed. Both the effects of
rotary inertia and shear must be included in the element model. The finite
element is assumed to be one dimensional of length L with the element
coordinate system centred at one end of it. The two nodes are defined in terms
of non-dimensional coordinates at ξ = 0, 1. The displacement and slope at
each node are the element degrees of freedom. The displacement is approximated
in terms of the shape functions Ni as

Hence
or otherwise derive the load distribution matrix due to a point load P acting
at the point ξ = ξP . Observe a few features one can exploit to
evaluate element matrices and set up the equations of motion for the synthesis
of feedback controls in MATLAB. Products in polynomials in the ξ can be
easily done by convolution in MATLAB using the conv.m m-file. The polynomials
may each be integrated in one dimension using the numerical quadrature m-file,
quad.m. Thus, one should first establish all the coefficients of the element
stiffness matrix prior to performing the integration as polynomials in
descending powers. (iv) Make suitable assumptions and generalize the element
matrices for a beam excited by piezoelectric patches. (v) Explain why it might
be useful to consider rotary inertia and shear effects when exciting the beam
with piezoelectric patches. Develop a finite element model of a cantilever beam
using 20 self-similar elements and 10 patches. Hence or otherwise develop a
model for exciting only the first three modes.