Engineers and scientists often use niass-spring models to gain insight into the dynamics of structures under the influence of disturbances such as earthquakes. Figure 1 below shows such a representation for a three-story building. For this case, the analysis is limited to horizontal motion of the structure. Using Newton's second law. force balances can be developed for this system as: Simulate the dynamics of this structure from t = 0 to 20 s. given the initial condition that the velocity of the ground floor is dx_1/dt = 1 m/s. and all other initial values of displacements and velocities are zero. Present your results as two timc-scries plots of: (i) displacements, and (ii) velocities. In addition, develop phase-plane plot of the displacements. Determine the eigen-values and eigenvectors and graphically represent the modes of vibration for the structure by displaying the amplitudes versus height for each of the eigenvectors. Normalize the amplitudes so that the displacement of the third floor is one