Predicting pitch jumps in bowed non-metallic plates

The vibration behavior of bowed metallic plates has been understood since Chladni’s pioneering work, but plates made of highly damped, non-metallic materials (paper, cardboard, plastic) reveal additional, less studied effects. When the edge of such a plate is bowed at an oblique angle, the heard pitch suddenly shifts at specific points along the bow stroke. The higher the plate’s fundamental frequency, the more often these jumps occur. Our measurements show that the phenomenon hinges on strong coupling between the bow hair and the plate. Whereas classic Chladni patterns are governed mainly by plate dynamics, the “pitch-jump” effect arises from repeated locking and freeing of the plate to three families of bow-hair modes: 1. modes spanning the entire bow-hair length, 2. ascending modes in the bow-hair segment ahead of the contact point, and 3. descending modes in the segment behind it. A graphical model is introduced that tracks and predicts these mode-locking cycles. Unlocking happens each time the locked mode’s node reaches the contact point, predicting both the positions and the growing density of pitch jumps observed in experiments.