Prob. 1 (30 Points); Given a 3-DOFs structure without damping, as shown below. Mass and stiffness matrices were formulated with small lateral deflection u1,u2, and u3. A dynamic load vector is applied to the structure.
Perform the following tasks in the orde r listed below (based on the material covered in Lectures 7 through 11) : (5 points deduction for Illegible HW)
a. Can the DOFs u1,u2,u3 be considered as generalized physical coordinates? Give some reasoning in support of your answer. b. Write the coupled equations of motion in physical coordinates (u1,u2,u3). c. Perform modal analysis; Using MATLAB function ‘eig’, compute eigenvalues and eigenvectors and graphically display the eigenvectors. d. Formulate Generalized Force in modal coordinates. The step forcing function F(z,t) is described above. e. Write the uncoupled E.O.M.s for forced response in modal coordinates. f. Write the closed form solution of each modal DOF. g. Transform modal response into physical response using modal superposition. h. Plot time history response of u1,u2,u3 over 0 to 2 secs and note where the maximum response occurs. i. Describe how you would compute shear force at each physical DOF by investigating the form of the stiffness matrix. Write the equation in matrix form.
( This problem is based on Problem 11, Chapter 8 of “Dynamics of Structures”, Hurty and Rubinstein)
