Mode Numbers
A mode of vibration is characterized by a modal frequency and a mode shape, and 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.
Read more about this topic: Normal Mode
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