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A normal mode of a dynamical 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 fixed frequencies.
In mathematics, the normal form of a dynamical system is a simplified form that can be useful in determining the system's behavior. Normal forms are often used for determining local bifurcations in a system. All systems exhibiting a certain type of bifurcation are locally (around the equilibrium) topologically equivalent to the normal form of ...
A system's normal mode is defined by the oscillation of a natural frequency in a sine waveform. In analysis of systems, it is convenient to use the angular frequency ω = 2 πf rather than the frequency f , or the complex frequency domain parameter s = σ + ω i .
Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular ...
Once a set of modes has been calculated for a system, the response to any kind of excitation can be calculated as a superposition of modes. This means that the response is the sum of the different mode shapes each one vibrating at its frequency. The weighting coefficients of this sum depend on the initial conditions and on the input signal.
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For example, normal modes of multidimensional harmonic oscillator (e.g. a system of beads arranged around the circle, connected with springs) corresponds to elementary vibrational modes of the system. In such a system zero modes typically occur and are related with a rigid rotation around the circle.
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