1.) 175 points] A rocket is clamped the launch pad like a vertical fixed-free beam (cantilever beam) as shown in the ill

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1 175 Points A Rocket Is Clamped The Launch Pad Like A Vertical Fixed Free Beam Cantilever Beam As Shown In The Ill 1 (68.23 KiB) Viewed 13 times

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: mium, m = 0.5m, m3 -0.25m. ka = k, k2 - 2k, ks = 2k. C1 = 0, C2 = 0, C3=0. F1 = 0, F2 = 0, F3 = 0. m3 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 = m m?/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 a = 1/m2, use the Characteristic Equation and the system of 3-equations and 3-unknowns to calculate the associated first mode shape vector (1) 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, 1.23, 1.04}" X (1) X(2) = (1.00, -0.05, -2.61)" X,(2) X(3) = {1.00, -8.19, 12.58}" X (3)