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. The 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 corresponding frequencies) that depend on its structure and composition.
The normal modes of a mechanical system are single frequency solutions to the equations of motion; the most general motion of the 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 many systems this is equivalent to reducing a collection of coupled oscillators to a set of decoupled, effective oscillators.
It is common to use a spring-mass system to illustrate a deformable structure. When such a system is excited at one of these natural frequencies, all of the masses move at the same frequency. The phases of the masses are the same, such that they all pass through both equilibrium and maximum amplitude simultaneously. The practical significance of this can be illustrated by a mass-spring model of a building. If an earthquake excites the system near one of the natural frequencies, the displacement of one floor with respect to another - depending on the mode - can be maximum. Obviously, buildings can only withstand this displacement up to a certain point. Modeling a building by finding its normal modes is an easy way to check the safety of the building's design. The concept of normal modes also finds application in wave theory, optics, quantum mechanics, and molecular dynamics.
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