Using Matlab and numerical integration, we can already see how the solution looks like. It looks kind of chaotic. Although all the variables have the same frequency, they may have different amplitudes. In the next section, we will see more explicitly how this superposition occurs. The mode frequencies and the mode amplitudes are the eigenvalues and eigenvectors of the eigenvalue problem associated with the system, respectively.
Normal modes are applicable to any linear system of springs and masses.
Thus, in particular, all the elements of the system must be linear. Under these conditions, the motion of a system of springs and masses evolves from the initial conditions as a superposition of the normal modes. A four state system Now let's return to a more complicated system with two masses and two springs. We will examine how to solve this system of equations analytically and how to relate that answer to the modes we talked about in the previous section. Note that although x is now a vector, the set of equations is still linear with constant coefficients.