MECHANICAL VIBRATIONS, VIBRATION SYNTHESIS, AND ANALYSIS Problem 1 Consider a uniform beam with the following boundary conditions at x=0: Clamped; At x=L the boundary is: guided end. Derive the first two modes of vibration analytically and graphically. Problem 2 Use the first function in Problem 1 to derive the single-term Galerkin approximation for the non-uniform or non-homogeneous beam assigned in Project 2. Conduct the two-term approximation Galerkin analysis and interpret the results. Compare with the exact solution for value Y=0, as described in lectures or as you can obtain with K functions. Problem 3 Conduct single-term Petrov-Galerkin method to problem 2 in this test, using (a) polynomial or trigonometric functions, and (b) exact solution for uniform beam.