Obtain the first and the second natural frequencies of the uniform cantilever of flexural stiffness.

Obtain the first and the second natural frequencies of the
uniform cantilever of flexural stiffness EI, length L and mass m, supported at
the free end by a spring of stiffness k and illustrated in Figure 3.16, by
applying the Rayleigh–Ritz method.

Plot the results as a function of the parameter β =
kL2/EI, 0 ≤ β ≤ 200 and comment on the influence of the
flexibility of the support. (ii) Assume that a distribution of displacement
sensors and PZT patch actuators are available and design a feedback controller
so that the closed-loop natural frequencies for the first two modes of a
uniform cantilever beam are equal to those of a beam with the same boundary
conditions as in part (i) and with the same length, mass and elastic
properties. State all the assumptions you have made in designing the
controller. (iii) Consider the behaviour of the closed-loop flexible beam
structure with the feedback controller in place and validate the performance of
the controller by considering at least the first four normal modes. (iv)
Discuss at least one application of the of the active controller designed and
validated in parts (ii) and (iii). 12. Consider a non-uniform beam of length L
and assume that a compressive longitudinal force P0 is acting at one end of the
beam. The beam is assumed to have a cross-sectional area A(x), flexural
rigidity EIzz (x) and material of density ρ(x), which are all functions of
the axial coordinate x along the beam, where the origin of the coordinate
system is located at the same end where the compressive force P0 is acting. The
transverse deflection of the beam along the beam axis is assumed to be w (x,
t). Assume that for a slender beam in transverse vibration, the stress and
strain are related according to the Bernoulli–Euler theory of bending. The beam
is acted on by a distributed axial loading q (x). The axial force acting at any
location is given by

Hence obtain the governing equation of motion at the instant
of buckling. (ii) Write the governing equation in terms of the non-dimensional
independent coordinate ξ = x/L, and assume that EIzz (x) and P (x) and the
mode of deflection η (x) are all polynomial functions of ξ (Li,
2009), given by

The recurrence relations are solved sequentially for βi
, i = n, n − 1, ··· , 1, 0. (iii) Assume that the boundary conditions at
particular end ξ = e could be one of four possibilities: (a) clamped (C:
η (e) = 0, dη dx ξ=e = 0); (b) hinged (H: η (e) = 0,
d2η dx 2 ξ=e = 0); (c) free (F: d2η dx 2 ξ=e = 0,
d3η dx 3 ξ=e = 0); (d) guided (G: dη dx ξ=e = 0,
d3η dx 3 ξ=e = 0). Show that the coefficients ai are given the
values listed in Table 3.1. Hence obtain the conditions for buckling. (iv) It
is proposed to strengthen the beams by actively restraining the beam in the
transverse direction at a finite number of locations. Assume a 20% increase in
the buckling load is desired in each case, and design and validate a suitable
distributed active controller. (Hint: Assume a distributed controller so the
closed-loop dynamics takes the form