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File:Drum vibration mode12.gif
Vibration of a single normal mode of a circular disc with a pinned boundary condition along the entire outer edge. See other modes.


File:A cup of black coffee vibrating in normal modes.jpeg
A flash photo of cup of black coffee vibrating in normal modes
File:Spherical harmonic in water drop.ogv
Excitation of normal modes in a drop of water during the Leidenfrost effect

In vibration, 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 motion described by the normal modes is called resonance. 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 that depend on its structure, materials and boundary conditions.

When relating to music, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics" or "overtones".

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.

The concept of normal modes also finds application in wave theory, optics, quantum mechanics, and molecular dynamics.

Mode numbersEdit

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 valley) it would be vibrating in mode 2.

In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using polar coordinates, we have a radial coordinate and an angular coordinate. If you measured from the center outward along the radial coordinate you would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the antisymmetric (also called skew-symmetry) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2-1 or 1-2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction).

Each mode is entirely independent of all other modes. Thus all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes.

NodesEdit

File:Mode Shape of a Round Plate with Node Lines.jpg
A mode shape of a drum membrane, with nodal lines shown in pale green.

In a one dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero. These nodes correspond to points in the mode shape where the mode shape is zero. Since the vibration of a system is given by the mode shape multiplied by a time function, the displacement of the node points remain zero at all times.

When expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles (one about half way between the edge and center, and the other on the edge itself) and a straight line bisecting the disk, where the displacement is close to zero. In a real system these lines would equal zero exactly, as shown to the right.


See also Edit


References Edit

  • Blevins, Robert D.. Formulas for natural frequency and mode shape.
  • Dynamics and Control of Distributed Systems.


External links Edit


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