5.13.3 Circular plate
By now you shouldn't be at all surprised to learn that when a circular plate that has an outer rim that is free to vibrate is struck, the plate will vibrate in a number of modes at the same time.
The first four modes of vibration of a circular plate with a free edge are shown in Figure 21. As with the circular membrane, the circles and lines correspond to nodal circles and nodal lines. Having said that, the dotted circle around the rim of the plate is not a nodal circle. In all its modes of vibration, the plate is free to vibrate at its edge. However, you shouldn't assume that the dotted circle is an antinode either; it simply indicates the boundary of the plate.
The fundamental vibrational mode is shown in Figure 21(a). There are two nodal lines through the centre dividing the plate into four segments. In this mode, two diametrically opposite quarters of the membrane vibrate in phase while the other two segments vibrate with the opposite phase. Again the angular position of these lines is determined by the place at which the plate is struck.
The second mode of vibration is shown in Figure 21(b). There is a single nodal circle located midway between the centre of the plate and the outside edge. The central disc vibrates with opposite phase to the outside ring. It rises as the outside ring falls and vice versa.
The third mode of vibration has six segments separated by three nodal lines, while the fourth mode of vibration contains one nodal line and one nodal circle.
Run the Flash animation below. It shows three-dimensional representations of a circular plate vibrating in each of its first four modes of vibration and should help you visualise the motion of the plate in each mode. Please note: to view this animation correctly, you will need to click on the ‘Launch in separate player’ link below.
Again, it turns out that the natural frequencies of a circular plate are not harmonically related. If the first natural frequency is denoted by f1 then the frequencies of the second, third and fourth modes of vibration are 1.73f1, 2.33f1 and 3.91f1 respectively. Again, you don't need to memorise these values. But do note that, once more, the natural frequencies are not harmonically related.