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ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag Date: 23 September 2014 Instructor: Lorenzo Dozio 1. A lightly damped, homogeneous, simply supported beam of length ℓ, bending stiffness EJ and mass per unit length m, is induced to vibrate at the supports by a prescribed sinusoidal ac celeration ¨wB at frequency ω 0. Assuming that the motion of the beam can be well represented by o nly its fundamental vibration mode, (a) compute the maximum value of |wC|, where wC is the vertical displacement at the center of the beam relative to the base; (b) compute the maximum value of the bending moment along the beam . wB wC EJ, m, ℓ Figure 1: Problem 1. 2. Let’s consider the system in Figure 2, consisting of a flexible cantile ver beam of length ℓ, bending stiffness EJ (x1) and mass per unit length m(x1), carrying a tip mass Mt and elastically connected through the spring K1 to a rigid bar of mass per unit length mR(x2) and length L. The rigid bar is also supported by the spring K2 and dashpot C. K1 K2 p0 Mt EJ, m, ℓ mR, L x1 x2 C Figure 2: Problem 2. The system is subjected to a distributed load p0(t)x2 acting on the rigid bar as an ergodic random process of power spectral density S(ω ) = ω 2 (1 + ω 2)(1 + 4ω 2) Assuming that the transverse displacement of the cantilever beam can be approximated as w(x1, t ) = (x1/ℓ )2u, write the state-space equations to compute the variance of the root bending moment of the cantilever beam. 3. Given the system ¨ z(t) = u(t), design a LQR controller to achieve a prescribed degree of stability α when Q = 02×2 and R = 1. Compute the closed-loop eigenvalues to check your design.