o o act im 1.) 175 points) A rocket is clamped the launch pad like a vertical fixed-free beam (cantilever beam) as shown

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O O Act Im 1 175 Points A Rocket Is Clamped The Launch Pad Like A Vertical Fixed Free Beam Cantilever Beam As Shown 1 (76.94 KiB) Viewed 12 times

o o act im 1.) 175 points) A rocket is clamped the launch pad like a vertical fixed-free beam (cantilever beam) as shown in the illustration. Analyze the structure to determine its longitudinal (long-axis/vertical) vibration characteristics. Consider the rocket to be a multi-degree of freedom system with three (3) degrees of freedom. Assume the following: m = m, m2 = 0.5m, m3 = 0.25m. kı = k, kz = 2k, k3 = 2k. C1 = 0, C2 = 0, C3 = 0. F1 = 0, F2 = 0, F3 = 0. ms a. [5 points) Create a diagram of the physical system with all associated mass, stiffness, damping, external force, and displacement elements. b. (20 points] Starting with the assumed values, write down the mass matrix (m) and the stiffness matrix [k]. Using the flexibility influence coefficients from example 6.5 compute the influence matrix (a). Use [m] and [a], compute the Dynamical Matrix [D] for the system. No need to write the [c] matrix. m2 C. [20 points] Using A = 12[1]-[D]I = 0, derive the Characteristic Determinant of the system. [Hint] it will be a 3 by 3 matrix in terms of a, where a=mo-/k. No need to compute the characteristic polynomial or its roots. d. (20 points] Given the first natural frequencies 01 = 0.73 (k/m)12, and recalling that 2 = 1/02, use the Characteristic Equation and the system of 3-equations and 3-unknowns to calculate the associated first mode shape vector X11). mi e. (10 points) Given the following 3 mode shape vectors, neatly and accurately plot each of the mode shapes based on the Eigenvectors. You can use the same graph for all three plots. [Note) vectors written in transposed notation. x(1) = {1.00, 1.231.04}" X (1) X12) = (1.00, -0.05, -2.61)' X,12) X(3) = {1.00, -8.19, 12.58}X2()