(i) Derive the frequency equation for the transverse vibration of a uniform beam fixed or clamped at one end and sliding at the other, as shown in Figure 3.14. (ii) Assume that a distribution of...

(i) Derive the frequency equation for the transverse vibration of a uniform beam fixed or clamped at one end and sliding at the other, as shown in Figure 3.14.

(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 active controller designed and validated in parts (ii) and (iii)