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ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag Date: 20 November 2014 Instructor: Lorenzo Dozio 1. Let’s consider the unrestrained composite rod in Figure 1 made by connecting two uniform rods together. ℓ1 ℓ2 x1 x2 E1, A 1, ρ 1 E2, A 2, ρ 2 Figure 1: Problem 1. (a) Write the exact boundary-value problem governing the free lon gitudinal vibrations of the system. (b) Write the corresponding eigenvalue problem and frequency equ ation. (c) Discuss the special case when the material of the two parts is t he same ( E1 = E2) and they have the same length ( ℓ1 = ℓ2). 2. A random vibrating table used to test some flying equipments is cap able of providing a zero mean value acceleration level with constant power spectral density of 0.02 g 2/Hz over the frequency range 10 − 1000 Hz. (a) Compute the root mean square value of the acceleration level o f the table. (b) Compute the root mean square value of the displacement level o f the table. 3. A homogeneous clamped-clamped beam is equipped with a collocated actuator-sensor pair in the middle in order to reduce the transverse vibration of the structure indu ced by a random distributed disturbance having uniform spatial variation and constant power spectral den sity. Show the equations and discuss the design of a frequency-shaped LQR control system incorpora ting an appropriate filter on the control input. Assume an ideal force actuator, an ideal accelerometer se nsor and a design model including only the fundamental model of the beam. [Describe all the steps and de fine all the quantities you need to solve the problem]