Estimating beam boundary stiffness using a physics-informed neural network
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Unknown boundary stiffness often complicates experimental vibration analysis. While beam-like setups are typically intended to approximate ideal “clamped” or “simply supported” conditions, achieving these in practice is nearly impossible. Even well-tightened beams exhibit flexibility at the boundaries, possibly making idealized assumptions insufficient. Consequently, discrepancies arise between experimental and theoretical vibrational responses—differences that may be irrelevant to the phenomenon under study but still hinder accurate interpretation. Since idealized boundary conditions cannot be obtained in practice, it would be valuable if easy-to-use methods existed for estimating the conditions of a specific setup, based on simple experimental measurements, so that the real conditions can be considered in analysis. However, estimating boundary stiffness through purely physics-based methods involves complex nonlinear regression and poses a challenging inverse problem [1,2]. Instead, we propose a physics-informed neural network (PINN) approach [3]. A cantilever-like beam serves as the case study. Using measured natural frequencies, the PINN predicts translational and rotational stiffness at the partially fixed end. A numerically generated dataset, spanning various stiffness combinations, maps the relationship between the first five natural frequencies and corresponding stiffness values. The physics-informed learner aims to minimize the residual of the beam’s transcendental frequency equation, ensuring consistency with governing mechanics. This approach offers two key advantages: low implementation complexity and meaningful integration of physics and data. The model requires simple algorithmic prescription and avoids intricate derivations, as physics are enforced in a forward manner. Performance depends on manual tuning of hyperparameters—such as network depth, learning rate, and the weighting of data and physics losses—but once optimized, the PINN achieves accurate stiffness estimation results. By combining experimental data with embedded physical laws, it may be possible to achieve a practical, fast, and scalable solution for estimating boundary conditions based on simple, practically obtainable natural frequency measurements.
