Modal Analysis of Axisymmetric Structures using Zernike Polynomials and Machine Learning

Abstract

The rise of electric mobility has amplified the need for advanced vibration analysis to control noise in both electric cars and aircraft. In vehicles, tire-induced vibrations have become a significant contributor to cabin noise, making an understanding of tire mode shapes crucial for effective vibration mitigation. Likewise, lightweight stiffened panels in electric aircraft demand careful vibration control to ensure passenger comfort. Addressing these challenges calls for innovative approaches not only to interpret complex vibration patterns but also to streamline the analysis process. In the first part, ML-based classification frameworks are developed for categorizing tire mode shapes, aiming to automate the traditionally manual and labor-intensive process. Leveraging Zernike Annular Moment Descriptors (ZAMD) as feature maps, supervised learning models, such as decision trees, random forests, and XGBoost, achieve a high classification accuracy, thus eliminating the need for manual intervention. Furthermore, convolutional neural networks (CNNs), trained on physics-informed modal displacement data from finite element analyses, are employed to classify tire mode shapes for both unloaded and loaded cases. The CNN-based approach, enhanced with transfer learning techniques, also achieves high classification accuracy, validating its effectiveness across different tire conditions. The second part of the thesis focuses on advancing vibration analysis methods for stiffened circular plates. The Ritz method, utilizing Zernike and Legendre polynomials as trial functions, is implemented to enable free-vibration analysis without the meshing constraints typically associated with traditional finite element methods. This approach allows arbitrary stiffener placement while maintaining computational efficiency and accuracy, particularly for higher-order modes. To address the limitations of the Ritz method in optimization studies, where large, fully populated matrices pose computational challenges, a graph neural network (GNN) model is proposed. The GNN, designed with edge-aware message passing, predicts the first natural frequency and corresponding Ritz constants for varying stiffener configurations, achieving low mean absolute errors on the test dataset. By integrating classical mathematical methods with modern machine learning techniques, this work presents a comprehensive framework for analyzing and interpreting free-vibration behavior in complex structural systems.