Consider the lateral vibrations of a string that is loaded with N different masses m that are located at different intervals L along the string. The left end of the string is fixed to a wall and the right end is secured in a vertical guide to ensure the string remains stretched. A uniform tension T is built up throughout the string. MN MN-1 m2 m L L u(x,t) u(x,t) UN(x,t) The vertical displacement of the i-th mass is denoted by ui(t). An external force fi(t) acts on each mass along u(t), including the end mass (at i = N). The forces fi(t) are not shown in the figure. The dashed horizontal line denotes the static equilibrium position of the string. The string vibrates with small amplitude about its equilibrium position. The mass of the string is negligible in comparison to m; so that you have as many degrees of freedom as the number of masses. The horizontal motion of the masses may be ignored, as well as energy losses (damping, friction, etc). Part I. Let u = [u(t), ..., un(t)] and obtain the governing equations for the system in the M+Ku = f matrix form: