Exam collections 2013

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ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag Date: 29 January 2013 Instructor: Lorenzo Dozio 1. Consider the homogeneous hollow cylindrical rod of lengt h ℓ, external radius Re and thickness h in Figure 1. The rod is made of aluminum and is connected to two end masses M1 and M2. The rod accommodates internally at x = αℓ (0 < α < 1) an undamped vibration absorber of mass ma and equivalent stiffness ka. x M1 M2 ma ka Re ℓ αℓ h F0 Figure 1: Hollow cylindrical rod embedding an undamped vibr ation absorber. (a) Derive the equations governing the exact free vibration problem. (b) Derive the equations governing the approximate free vibration problem when an admissible Ritz set is assumed for the longitudinal displacement of the rod as u(x, t ) = N(x)η(t) (explain your choice for N). (c) Derive the expression of the approximate frequency resp onse function H(jω ) = u(ℓ, jω ) F0(jω ) where F0 is a longitudinal force applied at the left end of the rod, usi ng the mode displacement method and the mode acceleration method. 2. Consider an ergodic random process described by ˙x(t) = −x(t) + n(t) where the input noise n(t) has zero mean value and autocovariance function knn(τ ) = e−2|τ | (a) Compute and plot kxx(τ ). (b) Compute σ 2 xx from the previous expression and check the result using the L yapunov equation. ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag Date: 26 June 2013 Instructor: Lorenzo Dozio 1. Consider the dynamic system in Figure 1, where the beam A-B of co nstant mass per unit length m is assumed to be rigid and the beam B-C of constant bending sti ffness EJ is assumed to be massless. The elastic beam is rigidly connected at point B and carries the tip point mas s M . The rigid beam is elastically supported by two springs of sti ffness k. w φ δ…