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ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag Date: 7 July 2014 Instructor: Lorenzo Dozio 1. Let us consider a homogeneous bar of length ℓ, torsional stiffness GJ and mass polar moment of inertia per unit length Ip. The bar is fixed at one end ( x = 0) and carries at the other end ( x = ℓ) a rigid disc of moment of inertia Id. (a) Determine the exact eigenfrequencies and eigenmodes of the s ystem. (b) Determine the orthogonality conditions of the eigenmodes. (c) Determine the approximate value of the fundamental frequen cy of the system by the Rayleigh’s quotient R using an appropriate static curve as trial function. (d) Show that R provides an upper bound for the lowest eigenfrequency of the sys tem under study. 2. Let us consider a homogeneous isotropic rectangular plate of sid e lengths a and b, thickness h, with simply supported edges. The plate is subjected to a transverse distribu ted random load p(t) of uniform spatial distribution p0 and covariance Cpp(τ) = e−a|τ | (a > 0). The plate is equipped at location ( x0, y 0) with an accelerometer and a band-limited inertial actuator modeled as an elastically suspended mass ma and internal dynamics expressed in the Laplace domain as δ(s) = b s + b δc(s) + F ka (b > 0) where δ is the elongation of the actuator, δc is the elongation due to the control, ka is the stiffness of the elastic support, and F is the force transmitted by the actuator to the plate. Write the state-space model of the system to be adopted for the design of a LQG control system aimed at minimizing the kinetic energy of the plate. Use a modal representa tion of the plate dynamics. 3. Let us consider a homogeneous rod of length ℓ, axial stiffness EA and mass per unit length m. The rod is fixed at one end ( x = 0) and carries a point mass of…