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