STUDY OF SIMILARITY PATTERN OF CHLADNI PLATE VIBRATIONS OF CIRCULAR AND SQUARE GEOMETRY USING SSIM

Abstract

Chladni patterns are formed on plates excited at resonant frequencies and can be observed through the arrangement of particles on the plate surface. This study investigates the structural similarity between numerical and experimental Chladni patterns for circular and square plates using the Structural Similarity Index Measure (SSIM). Numerical vibration mode shapes for circular plates, namely modes (0,1) and (0,2), were obtained by solving the Helmholtz equation using Bessel functions in polar coordinates. Square plate modes (2,1) and (2,2) were modeled using Fourier cosine approximations to the biharmonic plate equation under free-edge boundary conditions. Experimental images were obtained from the Physics Demonstration Archive of the University of California, Santa Barbara, and were preprocessed to match the simulated patterns in terms of grayscale representation, resolution and contrast. Image comparison was performed using Python with the NumPy, SciPy, OpenCV and scikit-image libraries. The SSIM values for corresponding modes indicate a high level of similarity, reaching 0.9326 and 0.9103 for circular plates, and 0.9026 and 0.7517 for square plates, indicating strong structural agreement. Cross-mode comparisons produced significantly lower similarity values for circular plates, but relatively higher values for square plates due to shared Cartesian symmetry. The results demonstrate that SSIM is an effective metric for validating theoretical vibration models and quantifying modal similarity, while providing a practical image-based approach for Chladni pattern analysis.