[Univ of Cambridge] [Dept of Engineering]

Chaldni Pattern Demo (Measurement of E)

The bending stiffness of a plate depends on the Young's modulus of the material, and the plate geometry. The vibration response of a plate is controlled by its bending stiffness - so plate vibration offers a route to the measurement of Young's modulus. In this video, rectangular and square plates are supported on small foam mounts over a loudspeaker, and the frequency adjusted until the plate resonates in one of its natural modes. The mode shape is revealed by a scattering of tea leaves on the plate - these move to the nodal lines, i.e. positions of zero displacement. The foam mounts need to be located under these lines, so as not to interfere with the out-of-plane displacement elsewhere.

For a rectangular plate, the lowest stiffness is for bending the plate in its longest direction - giving two nodal lines top to bottom. Increasing the frequency drives the next mode, bending about the stiffer shorter direction, giving horizontal nodal lines.

For a square plate, the bending stiffness is the same each way, and two different modes appear, to maintain the symmetry. The first illustrated produces a circular nodal line (with the centre moving in and out); the second (lower) frequency produces a corner-to-corner cross of nodal lines (after the sample is moved to locate the loudspeaker below the centre of one edge, where the displacement will be greatest).

                           
   Rectangular plate, first mode   Rectangular plate, second mode   Square plate, first mode   Square plate, second mode

Given the plate dimensions, the frequencies can be analytically related to the Young's modulus. But another benefit of this technique is that the Poisson's ratio influences the transition between successive modes. For the square plate, the ratio of the two frequencies illustrated here may be directly related to Poisson's ratio - the higher the ratio (i.e. the bigger the difference in pitch), the higher is Poisson's ratio. This is investigated using this apparatus in Part IB Experiment M1.

Listen carefully to the frequency changes in the video clip below.

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Last updated: January 2014
Dr Hugh Shercliff

hrs@eng.cam.ac.uk