Eigendrum: A Numerical Solver for Drumhead Vibrations and Sound Synthesis

How Eigendrum Works

Eigendrum determines the vibrational shapes and frequencies of a drumhead clamped at its rim. It addresses the eigenvalue problem −∇²u = λu with u = 0 on the edge, which typically lacks a formulaic solution for arbitrary shapes. The tool employs a numerical approach, meshing the drum shape with triangles to build finite element stiffness and mass matrices, then finding the smallest eigenvalues of Kφ = λMφ.

Accuracy and Validation

The numerical solver's accuracy is validated against shapes with known exact solutions, such as circles and rectangles. For these shapes, Eigendrum reproduces frequencies to better than a tenth of a percent. The conforming finite element method guarantees that its results are slight overestimates, ensuring reliability in the calculated numbers.

Impact of Strike Position

The location where a drumhead is struck directly influences the excitation of its vibrational modes. Striking a point where a mode has no displacement will not excite that specific mode. A single strike activates all modes simultaneously, with the mixture determined by the mallet's landing spot. Eigendrum also allows users to isolate and play individual modes, which is not possible with a physical mallet.

Designing Drum Shapes

Beyond tracing outlines, users can define drum shapes using mathematical equations. For example, r(t) = 1 + 0.3cos(5t) creates a five-lobed flower, and parametric x(t), y(t) pairs can generate complex closed curves like nephroids or eggs. This method enables the creation of precise and variable shapes, such as superellipses or multi-lobed figures, by adjusting numerical parameters. Shape definitions are stored directly in the URL, allowing for easy sharing and modification.