Two small rectangular steel parts, with identical masses of m 0 = 0.76 kg, are attached to the...

Two small rectangular steel parts, with identical masses of m 0 = 0.76 kg, are attached to the middle and the end of a slender steel cantilever beam as shown in Figure 3.18. The dimensions of the steel bar are given as l = 450 mm, b = 25.4 mm, and h = 5 mm. Young’s modulus (E) and the mass density (?) of the steel are E = 204 GPa and ? =7,860 kg/m 3 . Assume that half of the bar mass is added to the middle and a quarter is added to the end point of the beam, and assume that the assembly is approximated as a 2-DOF system when it vibrates in its most flexible direction x. Develop a comprehensive computer program to solve the following: a. Obtain the stiffness matrix in local (x) coordinates using the flexibility method. b. Derive the equation of motion in local coordinates using Newton’s law. Obtain the natural frequencies, modal matrix, and mode shapes. Plot the real mode shapes of the beam. c. Assume that the beam is modeled by two torsional springs at the base and at the middle. Using the torsional design stiffness of the beams at these points, express the equation of motion using the Lagrangian formulation. Use torsional displacements (? 1 ,? 2 ) as design coordinates. Solve for the natural frequencies and mode shapes. Comment on the differences between the solutions in local and design coordinates. d. Express the transformation matrix between the local (x) and design (?) coordinates. Obtain local mass and stiffness matrices using design stiffness, design mass, and coordinate transformation matrices. e. Express the modal mass and modal stiffness matrices. Predict the free vibrations of masses 1 and 2 when the tip of the bar is displaced 1 mm and released.