As already outlined in Chapter 1, modes of vibration arise as a natural concept in the study of the free vibrations of mechanical systems, discrete or continuous. In the continuous case, they are defined mathematically as many solutions of an eigenvalue problem involving the stiffness and mass operators introduced in Chapter 3. Provided the system is self-adjoint and statically stable, the discrete and infinite sequence of eigenvalues are positive and the related eigenvectors are real. From a physical viewpoint, they describe standing waves in which the material of the structure vibrates harmonically about a static and stable position of equilibrium, at specific frequencies and according to specific space shapes. Amongst several interesting properties, the most important one certainly is that the mode shapes can be used as an orthonormal vector basis to transform the partial derivative equations of motion into a set of time differential equations described by modal stiffness and mass matrices which operate on the so-called natural, or modal, coordinates of the material system. This new discretization procedure gives rise to a so called modal model, in which the response properties of the structure are characterized by a set of modal oscillators, instead of a set of finite elements. Many linear and even nonlinear problems of continuous mechanics are much more efficiently solved starting from a modal model, instead of a finite element model.