Mechanics-guided neural network surrogate model for cable force identification via single high-order frequency

Cable force is a critical parameter that significantly affects the safety and serviceability of cable-supported structures. However, directly measuring cable force presents challenges, particularly for in-service cables, due to the difficulty of installing force sensors. In practice, cable force is typically estimated using indirect vibration-based methods, which rely on the relationship between cable force and cable frequency and are often expressed through explicit, practical formulas. Nevertheless, a unified formula capable of accurately estimating cable force across various cables with different boundary conditions is still lacking. In this study, a mechanics-guided surrogate model is proposed to address this challenge using single high-order cable frequency. In particular, an inclined cable is fully characterized by three nondimensional parameters: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>λ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , and the sensitivity analysis in nondimensional space reveals that the mapping between high-order nondimensional frequency <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> is nearly single-valued and much simpler to capture. Leveraging this prior mechanical insight, we propose a neural network-based surrogate model, which operates in a well-designed nondimensional parameter space, to identify the cable force. By using nondimensional parameters for model training, the method achieves greater generalization and eliminates dependence on original physical parameters such as cable length. The proposed method has been validated via simulated, experimental, and real-world cables, demonstrating its capability to obtain accurate cable force identification with a single high-order frequency, especially performing well for short cables.