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ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag Date: 09 September 2014 Instructor: Lorenzo Dozio 1. The system in Figure 1 consists of a rigid hub with a cantilever flexible appendage carrying a tip payload. The hub is modeled as a rigid disk hinged at its center, with radius R and moment of inertia Ih. The appendage is modeled as a homogeneous slender beam of length ℓ, mass per unit length m and flexural stiffness EJ . The payload is modeled as a mass mt and inertia It. A prescribed torque Mc acts on the hub. x y X Y R, I h EJ, m, ℓ mt, I t Mc θ w Figure 1: Dynamic system of problem 1 (a) Write the exact linearized differential problem governing the planar dynamics of the s ystem (equations of motion and boundary conditions) in terms of rotation angle θ(t) and transverse deflection w(x, t ) of the flexible appendage. Assume small rotation speed ˙θ and small w. (b) Write the approximate equations of motion of the system according to an appropriate Ritz -Galerkin discretization and explain how to compute the bending moment along t he flexible appendage using the mode acceleration method. 2. The system in Figure 2 consists of a rigid body of mass M and moment of inertia J with respect to the center of mass (CM). The body is supported by two massless rods of length ℓ and damping coefficient β proportional to the stiffness. The sys- tem is forced by a prescribed base displacement w. Write the time-domain equations to compute the variance of the relative displacement z = v − w of CM when the base acceleration is a white noise of intensity W . w(t) v(t) EA, ℓ, β 2EA, ℓ, β M, J L1 L2 CM Figure 2: Dynamic system of problem 2 3. Let’s consider a LTI system ˙ z(t) = az(t) + bu(t), where the control variable u is actually provided by an actuator having an internal dynamics with…