ABSTRACT
The theory of oscillators, amplifiers, and deformations involves a linear second-order system with or without damping and forcing with one degree-of-freedom. It can be generalized in four directions: non-linear oscillations, associated with non-linear restoring or damping forces; higher-order systems, for example, the transverse deflection of one (two) dimensional elastic bodies with bending stiffness, including bars (plates); systems with several degrees-of-freedom, such as multidimensional oscillators with inertia, damping, resilience, and forcing; and systems with several independent variables, leading to partial differential equations. Very often, vibrating systems have more than one degree-of-freedom, for example, rigid bodies supported on springs and dampers, or associations of quasi-stationary electrical circuits; in most cases, these degrees-of-freedom interact and the multidimensional oscillator is described by a set of simultaneous ordinary differential equations. In order to illustrate the properties of N-dimensional oscillators with a minimum of algebra, it is sufficient to consider a coupled two-dimensional oscillator of which one example is the vibration absorber.
