In this experiment we studied the vibration of four different kinds of metal plates; circular, thick square, thin square and rectangular. To vibrate the plates we used a mechanical driver controlled by an electrical oscillator. The edges of the plates were unbound, while the source of vibration was located in the center of each plate. We were able to locate the different modal frequencies of each plate using an instrument that allowed us to delicately vary the frequency and amplitude of the driver. To locate various modal frequencies we sprinkled glass beads across the plate which would then settle along the various nodal lines at resonance. By counting the number of diametric and radial nodes (lines and circles) we could record the "m" and "n" values for each modal frequency. Finally, using the "m", "n" numbers and the frequency values for each mode of the plates we compared our data to either Chladni's Law or the general wave equation.
|The source of oscillation constantly produces different waves in the plate. However, at certain frequencies of oscillation standing waves are created in the plate by the continuous superposition of waves incident on and reflected from the edges. When this occurs the plate can be considered to be resonating at one of its natural frequencies. Each plate has many different frequencies, or modes of vibration, in which standing waves are created. The lowest mode of vibration is called the fundamental frequency and successive vibrational modes can be found by solving the two-dimensional wave equation. It is important to note that unlike standing waves in a string or in a column of air, the other vibrational modes of a plate are not integral multiples of the fundamental frequency and therefore do not form a harmonic series. In the vibration of plates the higher frequency modes are also quickly damped out and consequentially two-dimensional vibrations in flexible membranes, such as those in a drumhead, do not produce melodious sounds.|
At non-resonant frequencies the glass beads would dance erratically across the vibrating plate, however at the modal frequencies the glass beads would settle along the nodal lines. Since the edges of all our plates were unbound, they were free to oscillate at a maximum amplitude and are therefore considered antinodes of the standing waves created in the plates. Also since the center of each plate is at the driving source it is also considered an antinode. Throughout the rest of the plate, we encountered diametric and radial nodes and recorded their quantities at various nodal frequencies. While the number of diametric and radial nodes were difficult to count due to aberrations in the plates, our data did somewhat confirm the results predicted by the two-dimensional wave equation and by Chladni's Law.
while for circular plates, Chladni's Law predicts that the modal frequencies were approximately equal to
|We successfully determined the fundamental frequencies for all of our plates, however we were unable to locate each successive modal frequency. Some of the higher modal frequencies often yielded beautifully elaborate patterns and are particularly demonstrative of diametric and radial nodes. They are good examples of the presence of standing waves in plates that are vibrating at resonant frequencies.|
While the different vibrational modes of the plates are not integer multiples of their fundamental frequencies, the system of vibrating plates is still included in linear dynamics because there is only one possible maximum amplitude at the resonant frequency. The periodic driving force always produces a periodic output in the vibration of the plate, however only at resonance are standing waves created in the plate.