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DCASF A ID codes: 072652, 081225, 091932 Date: 30 June 2016 Instructor: Lorenzo Dozio 1. For the system in Figure 1, where u(x, t ) is the longitudinal displacement of the rod and z(t) is the absolute displacement of the mass M : (a) Derive the exact frequency equation when the axial stiffness EA and mass per unit length m of the rod are assumed to be constant and describe a numerical solution o f the above equation for computing the fundamental natural frequency of the problem. (b) Derive the approximate eigenvalue problem of the system modeled using a Ritz-Galerkin discre tization technique when EA = EA(x) and m = m(x) and describe a numerical method for computing the fundamental frequency of the discretized model. ℓ z(t) u(x, t ) M K1 K2 EA, m x Figure 1: A hybrid system consisting of a continuous rod connected to lumped elements. 2. The bending vibrations of a cantilever beam are modeled using a high -fidelity finite element representation using a lumped mass matrix and a mass-proportional damping model. S how how to compute the tip response to an imposed vertical displacement of the root in terms o f a quasi-static solution plus a modal solution. 3. Let’s consider an asymptotically stable LTI system with scalar inpu t u(t) and output y(t) assumed as ergodic random processes. (a) Prove that the power spectral density (PSD) function Syy (ω ) can be computed from the knowledge of the PSD function Suu(ω ) and the frequency response function H(jω ) of the system. (b) Compute H(s) when Syy = 1+2ω2 (1+ω2)(1+3ω2) and the input is a white noise of unit intensity. (c) Compute the variance σ 2 yy for the system of case (b). 4. Let’s consider a spring-mass-damper system, equipped with a fo rce actuator on the mass and a sensor measuring the acceleration of the mass. (a) Derive…