7. Systems with n degrees of freedom

7.1 General

Consider a mechanical system with n degrees of freedom, such that each of its oscillators can vibrate around an equilibrium position by reacting with neighboring oscillators. We need to define a number of position variables equal to the number of degrees of freedom of the system. In general, the free oscillations of this system are not harmonic. However, under certain conditions, the masses can execute harmonic oscillations at the same frequency, the masses being in phase or in opposition with respect to one of them taken as a reference. In this case, the system is said to have an eigenmode, and the oscillation frequency is an eigenfrequency. If the system vibrates according to an eigenmode, it is said to be normalized by setting the amplitude of one of the oscillators...