Identification of excitation frequencies of a vibrating beam using Dynamic Mode Decomposition

Identifying modal parameters for the different vibrational conditions for any structure or machine is very important for its structural health monitoring. Researchers have proposed different methodologies to determine that structure's excitation or natural frequency. In the initial design phase, researchers have to rely on building the FE model and then analyze its natural and excitation frequencies. However, for the complex structure, it is difficult to handle a huge dataset of the vibrational information of the mesh for multi-frequency excitations and it becomes computationally expensive to analyze the system's performance. This work presents the data-driven technique called Dynamic Mode Decomposition (DMD) which addresses this problem by discovering the system's dynamics using system's high-dimensional spatiotemporal data. The present work studies the FE vibration of fixed-fixed beams with single and multi-frequency harmonic forcing functions with and without noise for real-time scenarios. Here, DMD is applied to the sequential spatiotemporal data of displacement of the vibrating beams, and dominant frequencies and corresponding mode shapes are identified to predict the actual time dynamics of the system under different conditions in simulation and experimental conditions. This study also involves the failure of the DMD to capture the periodic oscillation (standing wave) in the beam and proposes the application of the Hankel matrix (time-delay coordinate) to capture the actual dynamical parameters of the system. The resulting predicted frequencies match the actual excitation frequencies, and the accuracy is improved with a number of stackings of the given time series data through the Hankel matrix at different rank truncations.