1. Find the first five natural frequencies and normalized mode shapes for a uniform bar, α(x) = 1, which is fixed at both ends, y(0) = 0 and y(1) = 0.
2. Find the first five natural frequencies and normalized mode shapes for a uniform bar, α(x) = 1, which is fixed at x = 0 and has a linear spring attached at x = 1 such that the boundary conditions are y(0) = 0 and dy/dx(1) + 0.25y(1) = 0.
3. Find the first five natural frequencies and normalized mode shapes for a uniform bar, α(x) = 1, which is fixed at x = 0 and has a discrete particle and a linear spring attached at x = 1 such that the boundary conditions are y(0) = 0 and dy/dx(1) + 0.25y(1) = 0.5ω2y(1).