Problem 4 The two springs as shown in the following figure have the same and constant stiffness value k. They can only deflect vertically, and in the equilibrium position the beam is horizontal. You are going to work on beam vibration problems with the assumption of small motions, constant mass per unit length and EI as well as uniform cross sectional area along the length of the beam and the following modifications. Please note that the beam becomes free- free in the limit k > 0, and pinned-pinned in the limit k co. U(x, x) K m, EI Show that the stiffness operator is symmetric and positive definite. x k [2 pts] Derive the characteristic equation and none-dimensionalize it in terms of the two parameters AL and k* (which will involve k, EI, and L and be a ratio between boundary and domain stiffness). [4 pts] Show that in the limits k* = 0 and co, the characteristic equation reduces to the ones for free and pinned beams. [4 pts] Write a program to find the natural frequencies as roots of the characteristic equation, and plot them as a function of k*. Plot the first several (between 3 to 6) natural frequencies over a range of k* that is large enough to show continuous transition from the free, to elastic, to pinned supports. A log scale would be helpful. Note how the lowest two modes of the pinned beam transition into the rigid body modes of the free beam. On your frequency diagram, next to each frequency locus, sketch by hand the shapes of the modes as k* changes.