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Consider the horizontal motion of the spring-mass system shown in the figure below. *2 13 k1 ka KA The horizontal deflections x1, x2 and x3 are measured relative to the position of static equilibrium. The spring stiffnesses, Ki,kark, and ke are the forces required to extend or compress each spring by unit length. a) Show that the equations of motion are mid = -kin + ka(x2 - x1) M2x2 = -K2(X2 - x1) + ka(X3 - x2) mgig = -Ka(X3 - X2) - K4X3 b) If the deflection vector is x = [x, x2 x3]', rewrite the equations of motion in the form * = Ax. c) Show that the substitution x = belt, where i = V-1, leads to the eigenvalue problem Ab = Ab, where 1= -off. The possible values that @ may assume are the natural circular frequencies of vibration of the system. d) Ifk, = K2 = kg = ka = 1 N/m, and m, = my = my = 1 kg, find the eigenvalues and eigenvectors of A using the following methods: (1) Manually solve det(4 - 2/) = 0 for the eigenvalues and Ab = Ab for the eigenvectors. Normalize the eigenvectors so that their Euclidean norm is one