Eigenmodes

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at the fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions.

The most general motion of a system is a superposition of its normal modes. The modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are orthogonal to each other.

In physics and engineering, for a dynamical system according to wave theory, a mode is a standing wave state of excitation, in which all the components of the system will be affected sinusoidally under a specified fixed frequency.

Because no real system can perfectly fit under the standing wave framework, the mode concept is taken as a general characterization of specific states of oscillation, thus treating the dynamic system in a linear fashion, in where linear superposition of states can be performed.

As classical examples, there are:

Most dynamical system can be excited under several modes. Each mode is characterized by one or several frequencies, according the modal variable field. For example, a vibrating rope in the 2D space is defined by a single-frequency (1D axial displacement), but a vibrating rope in the 3D space is defined by two frequencies -2D axial displacement-.

For a given amplitude on the modal variable, each mode will store a specific amount of energy, because of the sinusoidal excitation.

From all the modes of a dynamical system, the normal or dominant mode of a system, will be the mode storing the minimum amount of energy, for a given amplitude of the modal variable. Or equivalently, for a given stored amount of energy, will be the mode imposing the maximum amplitude of the modal variable.

A mode of vibration is characterized by a modal frequency and a mode shape. It is numbered according to the number of half waves in the vibration. For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave (one peak on the vibrating beam) it would be vibrating in mode 1. If it had a full sine wave (one peak and one trough) it would be vibrating in mode 2.

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