Exam collections 2012

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ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag Date: 26 June 2012 Instructor: Lorenzo Dozio 1. Consider the bar of length ℓ in Figure 1, with torsional stiffness GJ(x) = GJ = constant and mass polar moment of inertia per unit length Ip(x) = Ip = constant. The bar is clamped at one end and connected at the other end to a rigid disk by a torsional spring Ke. The disk, having mass moment of inertia ID and radius R, is connected to two massless strings stretched to a length L by a uniform tension T . x y z ID, R Ke GJ, I p, ℓ T, L T, L Figure 1: A uniform bar elastically connected to a restraine d disk. (a) Derive the equations governing the boundary-value prob lem. (b) Derive the characteristic equation governing the corre sponding eigenvalue problem. 2. Consider a cantilever beam of length ℓ, mass per unit length m(x) and flexural stiffness EJ (x) (see Figure 2). The beam is subjected to a random transverse load p er unit length p(x, t ) which is assumed to have a prescribed spatial distribution p0(x), zero mean value and power spectral density Spp(ω ) = a2 a2+ω2 (with a > 0). The beam is equipped with two displacement sensors and tw o ideal collocated force actuators at x = ℓ/ 2 and x = ℓ, respectively. The sensor-actuator pairs are used to imple ment an optimal direct velocity feedback (DVFB) controller for active damping aug mentation. x z p(x, t ) u1 u2 EJ (x), m (x), ℓ Figure 2: A cantilever beam equipped with two collocated sen sor-actuator pairs. (a) Derive a suitable reduced-order modal model of the beam d ynamics starting from an admissible Ritz-like approximation for the transverse displacement w(x, t ) = N(x)η(t) (explain your choice for N(x)). (b) Derive the state and output equations of the system to be u sed in the design of the DVFB controller.…